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Student "Basics" Package

The new Student Basics package helps to explore the foundations of higher math, making it possible to provide step-by-step breakdowns for expanding and simplifying mathematical expressions, such as simplifying fractions, expanding products of polynomials, or solving linear equations. All the steps to the solution are shown and documented, so that a student can easily understand what is happening at each stage of the solution. Students can use this package to understand where results are coming from and learn how to solve these problems on their own.

Here are 101 interesting examples showing the steps involved to solve or expand:

$with (Student :- Basics) &colon;$

$LinearSolveSteps (x + 3 = 8, x) &semi;$

$\begin{array}{c} x + 3 = 8 \\ x = 8 - 3 & (subtract from both sides) \\ x = 5 & (add terms) \end{array}$

(1)

$LinearSolveSteps (5 \cdot x - 2 = 13, x) &semi;$

$\begin{array}{c} 5 x - 2 = 13 \\ 5 x = 13 + 2 & (subtract from both sides) \\ x = \frac{13 + 2}{5} & (divide both sides) \\ x = \frac{15}{5} & (add terms) \\ x = 3 & (reduce fraction by gcd) \end{array}$

(2)

$LinearSolveSteps (4x + 5 = x - 4, x, implicitmultiply) &semi;$

$\begin{array}{c} 4 x + 5 = x - 4 \\ 4 x - x = −4 - 5 & (subtract from both sides) \\ 3 x = −4 - 5 & (add terms) \\ 3 x = −9 & (add terms) \\ x = \frac{−9}{3} & (divide both sides) \\ x = −3 & (reduce fraction by gcd) \end{array}$

(3)

$LinearSolveSteps (3(n - 1.8) = 2n + 1, n, implicitmultiply) &semi;$

$\begin{array}{c} 3 (n - 1.8) = 2 n + 1 \\ 3 n + 3 (−1.8) = 2 n + 1 & (distributive multiply) \\ 3 n - 5.4 = 2 n + 1 & (multiply constants) \\ 3 n - 2 n = 1 + 5.4 & (subtract from both sides) \\ n = 1 + 5.4 & (add terms) \\ n = 6.4 & (add terms) \end{array}$

(4)

$LinearSolveSteps (7y + 5 - 3y + 1 = 2y + 2, y, implicitmultiply) &semi;$

$\begin{array}{c} 7 y + 5 - 3 y + 1 = 2 y + 2 \\ 7 y - 3 y - 2 y = 2 - 5 - 1 & (subtract from both sides) \\ 2 y = 2 - 5 - 1 & (add terms) \\ 2 y = −4 & (add terms) \\ y = \frac{−4}{2} & (divide both sides) \\ y = −2 & (reduce fraction by gcd) \end{array}$

(5)

$LinearSolveSteps (2x + 4 = 10, x, implicitmultiply) &semi;$

$\begin{array}{c} 2 x + 4 = 10 \\ 2 x = 10 - 4 & (subtract from both sides) \\ x = \frac{10 - 4}{2} & (divide both sides) \\ x = \frac{6}{2} & (add terms) \\ x = 3 & (reduce fraction by gcd) \end{array}$

(6)

$LinearSolveSteps (3x - 4 = -10, x, implicitmultiply) &semi;$

$\begin{array}{c} 3 x - 4 = −10 \\ 3 x = −10 + 4 & (subtract from both sides) \\ x = \frac{−10 + 4}{3} & (divide both sides) \\ x = \frac{−6}{3} & (add terms) \\ x = −2 & (reduce fraction by gcd) \end{array}$

(7)

$LinearSolveSteps (4x - 4y = 8, x, implicitmultiply) &semi;$

$\begin{array}{c} 4 x - 4 y = 8 \\ 4 x = 8 + 4 y & (subtract from both sides) \\ x = \frac{8 + 4 y}{4} & (divide both sides) \\ x = \frac{4 (2 + y)}{4} & (factor) \\ x = 2 + y & (divide) \end{array}$

(8)

$LinearSolveSteps (x + 3^2 = 12, x) &semi;$

$\begin{array}{c} x + 3^{2} = 12 \\ x = 12 - 3^{2} & (subtract from both sides) \\ x = 12 - 9 & (evaluate power) \\ x = 3 & (add terms) \end{array}$

(9)

$LinearSolveSteps (x + y^{2} = 12, x) &semi;$

$\begin{array}{c} y^{2} + x = 12 \\ x = 12 - y^{2} & (subtract from both sides) \\ x = - y^{2} + 12 & (reorder terms) \end{array}$

(10)

$LinearSolveSteps (\frac{(x^{2} + y^{2})}{4} = \frac{1}{4} \cdot x^{2} - 2 \cdot x + 14, x)$

$\begin{array}{c} \frac{x^{2}}{4} + \frac{y^{2}}{4} = \frac{x^{2}}{4} + (−2) x + 14 \\ \frac{x^{2}}{4} - \frac{x^{2}}{4} - (−2) x = 14 - \frac{y^{2}}{4} & (subtract from both sides) \\ \frac{x^{2}}{4} - \frac{x^{2}}{4} + 2 x = 14 - \frac{y^{2}}{4} & (distribute negation) \\ 2 x = 14 - \frac{y^{2}}{4} & (add terms) \\ x = \frac{14 - \frac{y^{2}}{4}}{2} & (divide both sides) \end{array}$

(11)

$LinearSolveSteps (4(8-3x) = 32 - 8(x+2), x, implicitmultiply) &semi;$

$\begin{array}{c} 4 (8 - 3 x) = 32 - 8 (x + 2) \\ 4 \cdot 8 + 4 (- 3 x) = 32 - 8 (x + 2) & (distributive multiply) \\ 32 + 4 (- 3 x) = 32 - 8 (x + 2) & (multiply constants) \\ 32 + (−12) x = 32 - 8 (x + 2) & (multiply constants) \\ (−12) x + 8 (x + 2) = 32 - 32 & (subtract from both sides) \\ - 12 x + 8 x + 8 \cdot 2 = 32 - 32 & (distributive multiply) \\ - 12 x + 8 x + 16 = 32 - 32 & (multiply constants) \\ - 4 x + 16 = 32 - 32 & (add terms) \\ - 4 x + 16 = 0 & (add terms) \\ - 4 x = −16 & (subtract from both sides) \\ x = \frac{−16}{−4} & (divide both sides) \\ x = 4 & (reduce fraction by gcd) \end{array}$

(12)

$LinearSolveSteps (2x/3 = -18, x, implicitmultiply) &semi;$

$\begin{array}{c} \frac{2 x}{3} = −18 \\ 2 x = 3 (−18) & (multiply rhs by denominator of lhs) \\ x = \frac{3 (−18)}{2} & (divide both sides) \\ x = \frac{−54}{2} & (multiply constants) \\ x = −27 & (reduce fraction by gcd) \end{array}$

(13)

$LinearSolveSteps (2/(3x) = -18, x, implicitmultiply) &semi;$

$\begin{array}{c} \frac{2}{3 x} = −18 \\ \frac{1}{2} (3 x) = - \frac{1}{18} & (reciprocal of both sides) \\ x = \frac{- \frac{1}{18}}{\frac{3}{2}} & (divide both sides) \\ x = (- \frac{1}{18}) \frac{2}{3} & (rewrite division as multiplication by reciprocal) \\ x = - \frac{1}{27} & (multiply fraction and reduce by gcd) \end{array}$

(14)

$LinearSolveSteps ((2x)/(3x^2) = -18, x, implicitmultiply) &semi;$

$\begin{array}{c} \frac{2 x}{3 x^{2}} = −18 \\ \frac{2}{3 x} = −18 & (divide out common terms) \\ \frac{3 x}{2} = - \frac{1}{18} & (reciprocal of both sides) \\ x = \frac{- \frac{1}{18}}{\frac{3}{2}} & (divide both sides) \\ x = (- \frac{1}{18}) \frac{2}{3} & (rewrite division as multiplication by reciprocal) \\ x = - \frac{1}{27} & (multiply fraction and reduce by gcd) \end{array}$

(15)

$LinearSolveSteps (2/(3x) = -18, x, implicitmultiply) &semi;$

(16)

$LinearSolveSteps ((2x)/(3x^2) = -18, x, implicitmultiply) &semi;$

(17)

$LinearSolveSteps (x/3-5/12 = 3/4+1/2x, x, implicitmultiply) &semi;$

$\begin{array}{c} \frac{x}{3} - \frac{5}{12} = \frac{3}{4} + \frac{1}{2} x \\ \frac{x}{3} - \frac{1}{2} x = \frac{3}{4} + \frac{5}{12} & (subtract from both sides) \\ \frac{x}{3} - \frac{x}{2} = \frac{3}{4} + \frac{5}{12} & (multiply fraction) \\ - \frac{x}{6} = \frac{3}{4} + \frac{5}{12} & (add terms) \\ - \frac{x}{6} = \frac{7}{6} & (add terms) \\ - x = 6 \frac{7}{6} & (multiply rhs by denominator of lhs) \\ - x = \frac{42}{6} & (multiply fraction) \\ - x = 7 & (reduce fraction by gcd) \\ x = −7 & (negate both sides) \end{array}$

(18)

$LinearSolveSteps (10-3x = 7, x, implicitmultiply) &semi;$

$\begin{array}{c} 10 - 3 x = 7 \\ - 3 x = 7 - 10 & (subtract from both sides) \\ - 3 x = −3 & (add terms) \\ x = \frac{−3}{−3} & (divide both sides) \\ x = 1 & (divide out common terms) \end{array}$

(19)

$LinearSolveSteps (2(x+5)-7 = 3(x-2), x, implicitmultiply) &semi;$

$\begin{array}{c} 2 (x + 5) - 7 = 3 (x - 2) \\ 2 (x + 5) - 7 = 3 x + 3 (−2) & (distributive multiply) \\ 2 (x + 5) - 7 = 3 x - 6 & (multiply constants) \\ 2 (x + 5) - 3 x = −6 + 7 & (subtract from both sides) \\ 2 x + 2 \cdot 5 - 3 x = −6 + 7 & (distributive multiply) \\ 2 x + 10 - 3 x = −6 + 7 & (multiply constants) \\ - x + 10 = −6 + 7 & (add terms) \\ - x + 10 = 1 & (add terms) \\ - x = 1 - 10 & (subtract from both sides) \\ - x = −9 & (add terms) \\ x = 9 & (negate both sides) \end{array}$

(20)

$LinearSolveSteps (-x = 1+2, x) &semi;$

$\begin{array}{c} - x = 1 + 2 \\ - x = 3 & (add terms) \\ x = −3 & (negate both sides) \end{array}$

(21)

$LinearSolveSteps (-x = 1+y, x) &semi;$

$\begin{array}{c} - x = 1 + y \\ x = - (1 + y) & (negate both sides) \\ x = - 1 - y & (distribute negation) \end{array}$

(22)

$LinearSolveSteps (5x/4+1/2 = 2x-1/2, x, implicitmultiply) &semi;$

$\begin{array}{c} \frac{5 x}{4} + \frac{1}{2} = 2 x - \frac{1}{2} \\ \frac{5 x}{4} - 2 x = - \frac{1}{2} - \frac{1}{2} & (subtract from both sides) \\ - \frac{3 x}{4} = - \frac{1}{2} - \frac{1}{2} & (add terms) \\ - \frac{3 x}{4} = −1 & (add terms) \\ - 3 x = 4 (−1) & (multiply rhs by denominator of lhs) \\ - 3 x = −4 & (multiply constants) \\ x = \frac{−4}{−3} & (divide both sides) \\ x = \frac{4}{3} & (reduce fraction by gcd) \end{array}$

(23)

$LinearSolveSteps (.35y - .2 = .15y + .1, y, implicitmultiply) &semi;$

$\begin{array}{c} 0.35 y - 0.2 = 0.15 y + 0.1 \\ 0.35 y - 0.15 y = 0.1 + 0.2 & (subtract from both sides) \\ 0.20 y = 0.1 + 0.2 & (add terms) \\ 0.20 y = 0.3 & (add terms) \\ y = \frac{0.3}{0.20} & (divide both sides) \\ y = 1.500000000 & (divide constants) \end{array}$

(24)

$LinearSolveSteps (4x-1 = 4(x+3), x, implicitmultiply) &semi;$

$\begin{array}{c} 4 x - 1 = 4 (x + 3) \\ 4 x - 1 = 4 x + 4 \cdot 3 & (distributive multiply) \\ 4 x - 1 = 4 x + 12 & (multiply constants) \\ 4 x - 4 x = 12 + 1 & (subtract from both sides) \\ 0 = 12 + 1 & (add terms) \\ 0 = 13 & (add terms) \\ 0 = 13 & (no solution) \end{array}$

(25)

$LinearSolveSteps (5x+10 = 5(x+2), x, implicitmultiply) &semi;$

$\begin{array}{c} 5 x + 10 = 5 (x + 2) \\ 5 x + 10 = 5 x + 5 \cdot 2 & (distributive multiply) \\ 5 x + 10 = 5 x + 10 & (multiply constants) \\ 5 x - 5 x = 10 - 10 & (subtract from both sides) \\ 0 = 10 - 10 & (add terms) \\ 0 = 0 & (add terms) \\ 0 = 0 & (infinite number of solutions) \end{array}$

(26)

$LinearSolveSteps (xy + 6x = 1, x, implicitmultiply) &semi;$

$\begin{array}{c} x y + 6 x = 1 \\ x (6 + y) = 1 & (factor) \\ x = \frac{1}{6 + y} & (divide both sides) \end{array}$

(27)

$LinearSolveSteps ((x+1)/(2*y*z) = 4*y^2/z + 3*x/y, x) &semi;$

$\begin{array}{c} \frac{x + 1}{2 y z} = \frac{4 y^{2}}{z} + \frac{3 x}{y} \\ \frac{x + 1}{2 y z} - \frac{3 x}{y} = \frac{4 y^{2}}{z} & (subtract from both sides) \\ \frac{y (x + 1)}{2 y z y} + \frac{2 y z (- 3 x)}{2 y z y} = \frac{4 y^{2}}{z} & (find common denominator) \\ \frac{y (x + 1) + 2 y z (- 3 x)}{2 y z y} = \frac{4 y^{2}}{z} & (sum over common denominator) \\ \frac{y x + y \cdot 1 + 2 y z (- 3 x)}{2 y z y} = \frac{4 y^{2}}{z} & (distributive multiply) \\ \frac{x y + y + (−6) y z x}{2 y z y} = \frac{4 y^{2}}{z} & (multiply constants) \\ \frac{- 6 x y z + x y + y}{2 y z y} = \frac{4 y^{2}}{z} & (reorder terms) \\ \frac{y (- 6 x z + x + 1)}{y \cdot 2 y z} = \frac{4 y^{2}}{z} & (factor) \\ \frac{- 6 x z + x + 1}{2 y z} = \frac{4 y^{2}}{z} & (divide) \\ - 6 x z + x + 1 = 2 y z \frac{4 y^{2}}{z} & (multiply rhs by denominator of lhs) \\ - 6 x z + x = 2 y z \frac{4 y^{2}}{z} - 1 & (subtract from both sides) \\ - 6 x z + x = \frac{8 y^{3} z}{z} - 1 & (multiply fraction) \\ - 6 x z + x = 8 y^{3} - 1 & (divide) \\ x (1 - 6 z) = 8 y^{3} - 1 & (factor) \\ x = \frac{8 y^{3} - 1}{1 - 6 z} & (divide both sides) \end{array}$

(28)

$LinearSolveSteps (1/x = 3/4, x) &semi;$

$\begin{array}{c} \frac{1}{x} = \frac{3}{4} \\ x = \frac{4}{3} & (reciprocal of both sides) \end{array}$

(29)

$LinearSolveSteps (1/x = 4, x) &semi;$

$\begin{array}{c} \frac{1}{x} = 4 \\ x = \frac{1}{4} & (reciprocal of both sides) \end{array}$

(30)

$LinearSolveSteps (1/x - 1/2 = 3/4 - 2/x, x) &semi;$

$\begin{array}{c} \frac{1}{x} - \frac{1}{2} = \frac{3}{4} - \frac{2}{x} \\ \frac{1}{x} + \frac{2}{x} = \frac{3}{4} + \frac{1}{2} & (subtract from both sides) \\ \frac{3}{x} = \frac{3}{4} + \frac{1}{2} & (add terms) \\ \frac{3}{x} = \frac{5}{4} & (add terms) \\ \frac{x}{3} = \frac{4}{5} & (reciprocal of both sides) \\ x = \frac{\frac{4}{5}}{\frac{1}{3}} & (divide both sides) \\ x = \frac{4}{5} \frac{3}{1} & (rewrite division as multiplication by reciprocal) \\ x = \frac{12}{5} & (multiply fraction and reduce by gcd) \end{array}$

(31)

$LinearSolveSteps (3(n - 1.8) + 2(n-1) = 2(n + 1) - 3(n-2), n, implicitmultiply) &semi;$

$\begin{array}{c} 3 (n - 1.8) + 2 (n - 1) = 2 (n + 1) - 3 (n - 2) \\ 3 (n - 1.8) + 2 (n - 1) - 2 (n + 1) + 3 (n - 2) = 0 & (subtract from both sides) \\ 3 n + 3 (−1.8) + 2 (n - 1) - 2 (n + 1) + 3 (n - 2) = 0 & (distributive multiply) \\ 3 n - 5.4 + 2 (n - 1) - 2 (n + 1) + 3 (n - 2) = 0 & (multiply constants) \\ 3 n - 5.4 + 2 n + 2 (−1) - 2 (n + 1) + 3 (n - 2) = 0 & (distributive multiply) \\ 3 n - 5.4 + 2 n - 2 - 2 (n + 1) + 3 (n - 2) = 0 & (multiply constants) \\ 3 n - 5.4 + 2 n - 2 - (2 n + 2 \cdot 1) + 3 (n - 2) = 0 & (distributive multiply) \\ 3 n - 5.4 + 2 n - 2 - (2 n + 2) + 3 (n - 2) = 0 & (multiply constants) \\ 3 n - 5.4 + 2 n - 2 - (2 n + 2) + 3 n + 3 (−2) = 0 & (distributive multiply) \\ 3 n - 5.4 + 2 n - 2 - (2 n + 2) + 3 n - 6 = 0 & (multiply constants) \\ 6 n - 15.4 = 0 & (add terms) \\ 6 n = 15.4 & (subtract from both sides) \\ n = \frac{15.4}{6} & (divide both sides) \\ n = 2.566666667 & (divide constants) \end{array}$

(32)

$LinearSolveSteps (3(n - 1.8) + n*(2-1) = 2n + 1, n, implicitmultiply) &semi;$

$\begin{array}{c} 3 (n - 1.8) + n (2 - 1) = 2 n + 1 \\ 3 (n - 1.8) + n (2 - 1) - 2 n = 1 & (subtract from both sides) \\ 3 n + 3 (−1.8) + n (2 - 1) - 2 n = 1 & (distributive multiply) \\ 3 n - 5.4 + n (2 - 1) - 2 n = 1 & (multiply constants) \\ 3 n - 5.4 + n \cdot 1 - 2 n = 1 & (add terms) \\ 2 n - 5.4 = 1 & (add terms) \\ 2 n = 1 + 5.4 & (subtract from both sides) \\ 2 n = 6.4 & (add terms) \\ n = \frac{6.4}{2} & (divide both sides) \\ n = 3.200000000 & (divide constants) \end{array}$

(33)

$LinearSolveSteps (5x/4+1/2 = 2x-1/2, x, implicitmultiply) &semi;$

(34)

$LinearSolveSteps (x*(1-6*z) = 8*y-1, x) &semi;$

$\begin{array}{c} x (1 - 6 z) = 8 y - 1 \\ x = \frac{8 y - 1}{1 - 6 z} & (divide both sides) \end{array}$

(35)

$LinearSolveSteps (3*(x-6*z) = 8*y-1, x) &semi;$

$\begin{array}{c} 3 (x - 6 z) = 8 y - 1 \\ 3 x + 3 (- 6 z) = 8 y - 1 & (distributive multiply) \\ 3 x + (−18) z = 8 y - 1 & (multiply constants) \\ 3 x = 8 y - 1 - (−18) z & (subtract from both sides) \\ 3 x = 8 y - 1 + 18 z & (distribute negation) \\ x = \frac{8 y - 1 + 18 z}{3} & (divide both sides) \end{array}$

(36)

$LinearSolveSteps (10-3x = 7+2x, x, implicitmultiply) &semi;$

$\begin{array}{c} 10 - 3 x = 7 + 2 x \\ - 3 x - 2 x = 7 - 10 & (subtract from both sides) \\ - 5 x = 7 - 10 & (add terms) \\ - 5 x = −3 & (add terms) \\ x = \frac{−3}{−5} & (divide both sides) \\ x = \frac{3}{5} & (reduce fraction by gcd) \end{array}$

(37)

$LinearSolveSteps (10-3x = 7, x, implicitmultiply) &semi;$

(38)

$LinearSolveSteps (10 - 3 \cdot x = \frac{(7 + 3 \cdot x - y / 4)}{z}, x) &semi;$

$\begin{array}{c} 10 + (−3) x = \frac{7 + 3 x + (- \frac{1}{4}) y}{z} \\ (−3) x - \frac{7 + 3 x + (- \frac{1}{4}) y}{z} = −10 & (subtract from both sides) \\ - 3 x - \frac{7 + 3 x + \frac{- y}{4}}{z} = −10 & (multiply fraction) \\ \frac{z (- 3 x)}{z} + \frac{- (7 + 3 x + \frac{- y}{4})}{z} = −10 & (find common denominator) \\ \frac{z (- 3 x) - (7 + 3 x + \frac{- y}{4})}{z} = −10 & (sum over common denominator) \\ \frac{- 3 x z - 7 - 3 x + \frac{y}{4}}{z} = −10 & (distribute negation) \\ - 3 x z - 7 - 3 x + \frac{y}{4} = z (−10) & (multiply rhs by denominator of lhs) \\ - 3 x z - 3 x = z (−10) + 7 - \frac{y}{4} & (subtract from both sides) \\ - 3 x z - 3 x = 7 - 10 z - \frac{y}{4} & (reorder terms) \\ x ((−3) - 3 z) = 7 - 10 z - \frac{y}{4} & (factor) \\ x = \frac{7 - 10 z - \frac{y}{4}}{−3 - 3 z} & (divide both sides) \end{array}$

(39)

$LinearSolveSteps (\frac{1}{x + 2} = 1, x) &semi;$

$\begin{array}{c} {(x + 2)}^{−1} = 1 \\ x + 2 = 1 & (reciprocal of both sides) \\ x = 1 - 2 & (subtract from both sides) \\ x = −1 & (add terms) \end{array}$

(40)

$LinearSolveSteps (\frac{x}{x + 2} = 1, x) &semi;$

$\begin{array}{c} \frac{x}{x + 2} = 1 \\ x = (x + 2) \cdot 1 & (multiply rhs by denominator of lhs) \\ x = x + 2 & (multiply by 1) \\ x - x = 2 & (subtract from both sides) \\ 0 = 2 & (add terms) \\ 0 = 2 & (no solution) \end{array}$

(41)

$ExpandSteps ((3*a)*(4*a-y+42)) &semi;$

$\begin{array}{c} 3 a (4 a - y + 42) \\ = 3 a \cdot 4 a + 3 a (- y) + 3 a \cdot 42 & (distributive multiply) \\ = 12 a a + 3 a (- y) + 3 a \cdot 42 & (multiply constants) \\ = 12 a^{2} + 3 a (- y) + 3 a \cdot 42 & (multiply terms to exponential form) \\ = 12 a^{2} + (−3) a y + 3 a \cdot 42 & (multiply constants) \\ = 12 a^{2} - 3 a y + 126 a & (multiply constants) \end{array}$

(42)

$ExpandSteps ((3*a)*(4*a-y+42)) &semi;$

(43)

$ExpandSteps ((x^2)*(x^3)) &semi;$

$\begin{array}{c} x^{2} x^{3} \\ = x^{5} & (add exponents with common base) \end{array}$

(44)

$ExpandSteps ((x^2*y/(x*y))) &semi;$

$\begin{array}{c} \frac{x^{2} y}{x y} \\ = \frac{x^{2}}{x} & (divide out common terms) \\ = x & (divide) \end{array}$

(45)

$ExpandSteps ((2*x^2*y/(4*x*y))) &semi;$

$\begin{array}{c} \frac{2 x^{2} y}{4 x y} \\ = \frac{2 x^{2}}{4 x} & (divide out common terms) \\ = \frac{2 x}{4} & (divide out common terms) \\ = \frac{x}{2} & (reduce fraction by gcd) \end{array}$

(46)

$ExpandSteps ((2.1*x)/4.3) &semi;$

$\begin{array}{c} \frac{2.1 x}{4.3} \\ = 0.4883720930 x & (divide constants) \end{array}$

(47)

$ExpandSteps ((2.1*x^2*y/(4.3*x*y))) &semi;$

$\begin{array}{c} \frac{2.1 x^{2} y}{4.3 x y} \\ = \frac{2.1 x^{2}}{4.3 x} & (divide out common terms) \\ = \frac{2.100000000 x}{4.3} & (divide out common terms) \\ = 0.4883720930 x & (divide constants) \end{array}$

(48)

$ExpandSteps ((x^2*y+y^2⋅x)/(x+y)) &semi;$

$\begin{array}{c} \frac{x^{2} y + y^{2} x}{x + y} \\ = \frac{(x + y) x y}{x + y} & (factor) \\ = x y & (divide) \end{array}$

(49)

$ExpandSteps ((-y)^2) &semi;$

$\begin{array}{c} {(- y)}^{2} \\ = {(−1)}^{2} y^{2} & (distribute exponent to individual terms) \\ = 1 y^{2} & (evaluate power) \end{array}$

(50)

$ExpandSteps ((-y^2)+y^2) &semi;$

$\begin{array}{c} - y^{2} + y^{2} \\ = 0 & (add terms) \end{array}$

(51)

$ExpandSteps ((x^2-y^2)/(x+y)) &semi;$

$\begin{array}{c} \frac{x^{2} - y^{2}}{x + y} \\ = \frac{(x + y) (x - y)}{x + y} & (factor) \\ = x - y & (divide) \end{array}$

(52)

$ExpandSteps (2*(-y^2)) &semi;$

$\begin{array}{c} 2 (- y^{2}) \\ = (−2) y^{2} & (multiply constants) \end{array}$

(53)

$ExpandSteps (2*(x^2-y^2)) &semi;$

$\begin{array}{c} 2 (x^{2} - y^{2}) \\ = 2 x^{2} + 2 (- y^{2}) & (distributive multiply) \\ = 2 x^{2} + (−2) y^{2} & (multiply constants) \end{array}$

(54)

$ExpandSteps ((2*(x^2-y^2))/(4*(x+y))) &semi;$

$\begin{array}{c} \frac{2 (x^{2} - y^{2})}{4 (x + y)} \\ = \frac{2 x^{2} + 2 (- y^{2})}{4 (x + y)} & (distributive multiply) \\ = \frac{2 x^{2} + (−2) y^{2}}{4 (x + y)} & (multiply constants) \\ = \frac{2 x^{2} - 2 y^{2}}{4 x + 4 y} & (distributive multiply) \\ = \frac{(2 x + 2 y) (x - y)}{(2 x + 2 y) \cdot 2} & (factor) \\ = \frac{x - y}{2} & (divide) \end{array}$

(55)

$ExpandSteps ((2.1*(x^2-y^2))/4) &semi;$

$\begin{array}{c} \frac{2.1 (x^{2} - y^{2})}{4} \\ = \frac{2.1 x^{2} + 2.1 (- y^{2})}{4} & (distributive multiply) \\ = \frac{2.1 x^{2} + (−2.1) y^{2}}{4} & (multiply constants) \\ = 0.5250000000 x^{2} - 0.5250000000 y^{2} & (divide constants) \end{array}$

(56)

$ExpandSteps ((2.1*(x^2-y^2))/(4*(x+y))) &semi;$

$\begin{array}{c} \frac{2.1 (x^{2} - y^{2})}{4 (x + y)} \\ = \frac{2.1 x^{2} + 2.1 (- y^{2})}{4 (x + y)} & (distributive multiply) \\ = \frac{2.1 x^{2} + (−2.1) y^{2}}{4 (x + y)} & (multiply constants) \\ = \frac{2.1 x^{2} - 2.1 y^{2}}{4 x + 4 y} & (distributive multiply) \\ = \frac{(x + y) (2.100000000 x - 2.100000000 y)}{(x + y) \cdot 4.} & (factor) \\ = \frac{2.100000000 x - 2.100000000 y}{4.} & (divide) \\ = 0.5250000000 x - 0.5250000000 y & (divide constants) \end{array}$

(57)

$ExpandSteps ((x^2/z)*(z^3/x^2)) &semi;$

$\begin{array}{c} \frac{x^{2}}{z} \frac{z^{3}}{x^{2}} \\ = \frac{x^{2} z^{3}}{z x^{2}} & (multiply fraction) \\ = \frac{x^{2} z^{2}}{x^{2}} & (divide out common terms) \\ = z^{2} & (divide out common terms) \end{array}$

(58)

$ExpandSteps ((17*x^4*y^2/(64*z^5)) * (24*y*z^2/(85*x^2))) &semi;$

$\begin{array}{c} \frac{17 x^{4} y^{2}}{64 z^{5}} \frac{24 y z^{2}}{85 x^{2}} \\ = \frac{408 x^{4} y^{3} z^{2}}{5440 z^{5} x^{2}} & (multiply fraction) \\ = \frac{408 x^{4} y^{3}}{5440 x^{2} z^{3}} & (divide out common terms) \\ = \frac{408 x^{2} y^{3}}{5440 z^{3}} & (divide out common terms) \\ = \frac{3 x^{2} y^{3}}{40 z^{3}} & (reduce fraction by gcd) \end{array}$

(59)

$ExpandSteps (3^2) &semi;$

$\begin{array}{c} 3^{2} \\ = 9 & (evaluate power) \end{array}$

(60)

$ExpandSteps (`%+` (`%+` (9 &ast; a &Hat; 2, 6 &ast; a &ast; b), `%+` (6 &ast; a &ast; b, 4 &ast; b &Hat; 2))) &semi;$

$\begin{array}{c} 9 a^{2} + 6 a b + 6 a b + 4 b^{2} \\ = 9 a^{2} + 12 a b + 4 b^{2} & (add terms) \end{array}$

(61)

$ExpandSteps ((3*a+2*b)^2) &semi;$

$\begin{array}{c} {(3 a + 2 b)}^{2} \\ = (3 a + 2 b) (3 a + 2 b) & (rewrite exponentiation as multiplication) \\ = 3 a (3 a + 2 b) + 2 b (3 a + 2 b) & (distributive multiply) \\ = 3 a \cdot 3 a + 3 a \cdot 2 b + 2 b (3 a + 2 b) & (distributive multiply) \\ = 9 a a + 3 a \cdot 2 b + 2 b (3 a + 2 b) & (multiply constants) \\ = 9 a^{2} + 3 a \cdot 2 b + 2 b (3 a + 2 b) & (multiply terms to exponential form) \\ = 9 a^{2} + 6 a b + 2 b (3 a + 2 b) & (multiply constants) \\ = 9 a^{2} + 6 a b + 2 b \cdot 3 a + 2 b \cdot 2 b & (distributive multiply) \\ = 9 a^{2} + 6 a b + 6 b a + 2 b \cdot 2 b & (multiply constants) \\ = 9 a^{2} + 6 a b + 6 a b + 4 b b & (multiply constants) \\ = 9 a^{2} + 6 a b + 6 a b + 4 b^{2} & (multiply terms to exponential form) \\ = 9 a^{2} + 12 a b + 4 b^{2} & (add terms) \end{array}$

(62)

$ExpandSteps ((3a+2b)*(4a-y+42), implicitmultiply) &semi;$

$\begin{array}{c} (3 a + 2 b) (4 a - y + 42) \\ = 3 a (4 a - y + 42) + 2 b (4 a - y + 42) & (distributive multiply) \\ = 3 a \cdot 4 a + 3 a (- y) + 3 a \cdot 42 + 2 b (4 a - y + 42) & (distributive multiply) \\ = 12 a a + 3 a (- y) + 3 a \cdot 42 + 2 b (4 a - y + 42) & (multiply constants) \\ = 12 a^{2} + 3 a (- y) + 3 a \cdot 42 + 2 b (4 a - y + 42) & (multiply terms to exponential form) \\ = 12 a^{2} + (−3) a y + 3 a \cdot 42 + 2 b (4 a - y + 42) & (multiply constants) \\ = 12 a^{2} - 3 a y + 126 a + 2 b (4 a - y + 42) & (multiply constants) \\ = 12 a^{2} - 3 a y + 126 a + 2 b \cdot 4 a + 2 b (- y) + 2 b \cdot 42 & (distributive multiply) \\ = 12 a^{2} - 3 a y + 126 a + 8 b a + 2 b (- y) + 2 b \cdot 42 & (multiply constants) \\ = 12 a^{2} - 3 a y + 126 a + 8 a b + (−2) b y + 2 b \cdot 42 & (multiply constants) \\ = 12 a^{2} - 3 a y + 126 a + 8 a b - 2 b y + 84 b & (multiply constants) \\ = 12 a^{2} + 8 a b - 3 a y - 2 b y + 126 a + 84 b & (reorder terms) \end{array}$

(63)

$ExpandSteps (3*3) &semi;$

$\begin{array}{c} 3 \cdot 3 \\ = 9 & (multiply constants) \end{array}$

(64)

$ExpandSteps (1*2*3*4*5*6*7*8*9) &semi;$

$\begin{array}{c} 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \\ = 362880 & (multiply constants) \end{array}$

(65)

$ExpandSteps (1+1) &semi;$

$\begin{array}{c} 1 + 1 \\ = 2 & (add terms) \end{array}$

(66)

$ExpandSteps (0^x) &semi;$

$\begin{array}{c} 0^{x} \\ = 0 & (evaluate power) \end{array}$

(67)

$ExpandSteps (x^0) &semi;$

$\begin{array}{c} x^{0} \\ = 1 & (x^0 = 1) \end{array}$

(68)

$ExpandSteps (5^0) &semi;$

$\begin{array}{c} 5^{0} \\ = 1 & (x^0 = 1) \end{array}$

(69)

$ExpandSteps ((a*b)^3) &semi;$

$\begin{array}{c} {(a b)}^{3} \\ = a^{3} b^{3} & (distribute exponent to individual terms) \end{array}$

(70)

$ExpandSteps (a^3*a^2) &semi;$

$\begin{array}{c} a^{3} a^{2} \\ = a^{5} & (add exponents with common base) \end{array}$

(71)

$ExpandSteps ((a+b)^2) &semi;$

$\begin{array}{c} {(a + b)}^{2} \\ = (a + b) (a + b) & (rewrite exponentiation as multiplication) \\ = a (a + b) + b (a + b) & (distributive multiply) \\ = a a + a b + b (a + b) & (distributive multiply) \\ = a^{2} + a b + b (a + b) & (multiply terms to exponential form) \\ = a^{2} + a b + b a + b b & (distributive multiply) \\ = a^{2} + a b + a b + b^{2} & (multiply terms to exponential form) \\ = a^{2} + 2 a b + b^{2} & (add terms) \end{array}$

(72)

$ExpandSteps ({(a + b)}^{5}) &semi;$

$\begin{array}{c} {(a + b)}^{5} \\ = (a + b) (a + b) (a + b) (a + b) (a + b) & (rewrite exponentiation as multiplication) \\ = (a (a + b) + b (a + b)) (a + b) (a + b) (a + b) & (distributive multiply) \\ = (a a + a b + b (a + b)) (a + b) (a + b) (a + b) & (distributive multiply) \\ = (a^{2} + a b + b (a + b)) (a + b) (a + b) (a + b) & (multiply terms to exponential form) \\ = (a^{2} + a b + b a + b b) (a + b) (a + b) (a + b) & (distributive multiply) \\ = (a^{2} + a b + a b + b^{2}) (a + b) (a + b) (a + b) & (multiply terms to exponential form) \\ = (a^{2} + 2 a b + b^{2}) (a + b) (a + b) (a + b) & (add terms) \\ = ((a^{2} + 2 a b + b^{2}) a + (a^{2} + 2 a b + b^{2}) b) (a + b) (a + b) & (distributive multiply) \\ = (a a^{2} + a \cdot 2 a b + a b^{2} + (a^{2} + 2 a b + b^{2}) b) (a + b) (a + b) & (distributive multiply) \\ = (a^{3} + a \cdot 2 a b + a b^{2} + (a^{2} + 2 a b + b^{2}) b) (a + b) (a + b) & (add exponents with common base) \\ = (a^{3} + 2 a^{2} b + a b^{2} + (a^{2} + 2 a b + b^{2}) b) (a + b) (a + b) & (multiply terms to exponential form) \\ = (a^{3} + 2 a^{2} b + a b^{2} + b a^{2} + b \cdot 2 a b + b b^{2}) (a + b) (a + b) & (distributive multiply) \\ = (a^{3} + 2 a^{2} b + a b^{2} + a^{2} b + 2 b^{2} a + b b^{2}) (a + b) (a + b) & (multiply terms to exponential form) \\ = (a^{3} + 2 a^{2} b + a b^{2} + a^{2} b + 2 a b^{2} + b^{3}) (a + b) (a + b) & (add exponents with common base) \\ = (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}) (a + b) (a + b) & (add terms) \\ = ((a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}) a + (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}) b) (a + b) & (distributive multiply) \\ = (a a^{3} + a \cdot 3 a^{2} b + a \cdot 3 a b^{2} + a b^{3} + (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}) b) (a + b) & (distributive multiply) \\ = (a^{4} + a \cdot 3 a^{2} b + a \cdot 3 a b^{2} + a b^{3} + (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}) b) (a + b) & (add exponents with common base) \\ = (a^{4} + 3 a^{3} b + a \cdot 3 a b^{2} + a b^{3} + (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}) b) (a + b) & (add exponents with common base) \\ = (a^{4} + 3 a^{3} b + 3 a^{2} b^{2} + a b^{3} + (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}) b) (a + b) & (multiply terms to exponential form) \\ = (a^{4} + 3 a^{3} b + 3 a^{2} b^{2} + a b^{3} + b a^{3} + b \cdot 3 a^{2} b + b \cdot 3 a b^{2} + b b^{3}) (a + b) & (distributive multiply) \\ = (a^{4} + 3 a^{3} b + 3 a^{2} b^{2} + a b^{3} + a^{3} b + 3 b^{2} a^{2} + b \cdot 3 a b^{2} + b b^{3}) (a + b) & (multiply terms to exponential form) \\ = (a^{4} + 3 a^{3} b + 3 a^{2} b^{2} + a b^{3} + a^{3} b + 3 a^{2} b^{2} + 3 b^{3} a + b b^{3}) (a + b) & (add exponents with common base) \\ = (a^{4} + 3 a^{3} b + 3 a^{2} b^{2} + a b^{3} + a^{3} b + 3 a^{2} b^{2} + 3 a b^{3} + b^{4}) (a + b) & (add exponents with common base) \\ = (a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}) (a + b) & (add terms) \\ = (a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}) a + (a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}) b & (distributive multiply) \\ = a a^{4} + a \cdot 4 a^{3} b + a \cdot 6 a^{2} b^{2} + a \cdot 4 a b^{3} + a b^{4} + (a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}) b & (distributive multiply) \\ = a^{5} + a \cdot 4 a^{3} b + a \cdot 6 a^{2} b^{2} + a \cdot 4 a b^{3} + a b^{4} + (a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}) b & (add exponents with common base) \\ = a^{5} + 4 a^{4} b + a \cdot 6 a^{2} b^{2} + a \cdot 4 a b^{3} + a b^{4} + (a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}) b & (add exponents with common base) \\ = a^{5} + 4 a^{4} b + 6 a^{3} b^{2} + a \cdot 4 a b^{3} + a b^{4} + (a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}) b & (add exponents with common base) \\ = a^{5} + 4 a^{4} b + 6 a^{3} b^{2} + 4 a^{2} b^{3} + a b^{4} + (a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}) b & (multiply terms to exponential form) \\ = a^{5} + 4 a^{4} b + 6 a^{3} b^{2} + 4 a^{2} b^{3} + a b^{4} + b a^{4} + b \cdot 4 a^{3} b + b \cdot 6 a^{2} b^{2} + b \cdot 4 a b^{3} + b b^{4} & (distributive multiply) \\ = a^{5} + 4 a^{4} b + 6 a^{3} b^{2} + 4 a^{2} b^{3} + a b^{4} + a^{4} b + 4 b^{2} a^{3} + b \cdot 6 a^{2} b^{2} + b \cdot 4 a b^{3} + b b^{4} & (multiply terms to exponential form) \\ = a^{5} + 4 a^{4} b + 6 a^{3} b^{2} + 4 a^{2} b^{3} + a b^{4} + a^{4} b + 4 a^{3} b^{2} + 6 b^{3} a^{2} + b \cdot 4 a b^{3} + b b^{4} & (add exponents with common base) \\ = a^{5} + 4 a^{4} b + 6 a^{3} b^{2} + 4 a^{2} b^{3} + a b^{4} + a^{4} b + 4 a^{3} b^{2} + 6 a^{2} b^{3} + 4 b^{4} a + b b^{4} & (add exponents with common base) \\ = a^{5} + 4 a^{4} b + 6 a^{3} b^{2} + 4 a^{2} b^{3} + a b^{4} + a^{4} b + 4 a^{3} b^{2} + 6 a^{2} b^{3} + 4 a b^{4} + b^{5} & (add exponents with common base) \\ = a^{5} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + b^{5} & (add terms) \end{array}$

(73)

Note that this could be expanded but the system chooses not to as the output would be excessively large (the cut-off is an exponent $\geq$ 100)

$ExpandSteps ({(a + b)}^{1000}) &semi;$

$\begin{array}{c} {(a + b)}^{1000} \end{array}$

(74)

$ExpandSteps ((a+b)^(1/2) ⋅ (a+b)^(3/2)) &semi;$

$\begin{array}{c} {(a + b)}^{\frac{1}{2}} {(a + b)}^{\frac{3}{2}} \\ = {(a + b)}^{2} & (add exponents with common base) \\ = (a + b) (a + b) & (rewrite exponentiation as multiplication) \\ = a (a + b) + b (a + b) & (distributive multiply) \\ = a a + a b + b (a + b) & (distributive multiply) \\ = a^{2} + a b + b (a + b) & (multiply terms to exponential form) \\ = a^{2} + a b + b a + b b & (distributive multiply) \\ = a^{2} + a b + a b + b^{2} & (multiply terms to exponential form) \\ = a^{2} + 2 a b + b^{2} & (add terms) \end{array}$

(75)

$ExpandSteps ((1+I)^1.5) &semi;$

$\begin{array}{c} {(1 + I)}^{1.5} \\ = {(1 + I)}^{1.5} & (add terms) \\ = 0.6435942529 + 1.553773974 \cdot I & (evaluate power) \end{array}$

(76)

$ExpandSteps (a/(2*b/a)) &semi;$

$\begin{array}{c} \frac{a}{\frac{2 b}{a}} \\ = a \frac{a}{2 b} & (rewrite division as multiplication by reciprocal) \\ = \frac{a^{2}}{2 b} & (multiply fraction and reduce by gcd) \end{array}$

(77)

$ExpandSteps (a/(2*a)) &semi;$

$\begin{array}{c} \frac{a}{2 a} \\ = \frac{1}{2} & (divide out common terms) \end{array}$

(78)

$ExpandSteps (3a/(6a), implicitmultiply) &semi;$

$\begin{array}{c} \frac{3 a}{6 a} \\ = \frac{3}{6} & (divide out common terms) \\ = \frac{1}{2} & (reduce fraction by gcd) \end{array}$

(79)

$ExpandSteps ((3x sin(x))/x, implicitmultiply) &semi;$

$\begin{array}{c} \frac{3 x \sin (x)}{x} \\ = 3 \sin (x) & (divide out common terms) \end{array}$

(80)

$ExpandSteps (3( sin(x) + y ), implicitmultiply) &semi;$

$\begin{array}{c} 3 (\sin (x) + y) \\ = 3 \sin (x) + 3 y & (distributive multiply) \end{array}$

(81)

$ExpandSteps ((3a+2b)*(4a-y^2+42), implicitmultiply) &semi;$

$\begin{array}{c} (3 a + 2 b) (4 a - y^{2} + 42) \\ = (3 a + 2 b) (- y^{2} + 4 a + 42) & (reorder terms) \\ = (- y^{2} + 4 a + 42) \cdot 3 a + (- y^{2} + 4 a + 42) \cdot 2 b & (distributive multiply) \\ = 3 a (- y^{2}) + 3 a \cdot 4 a + 3 a \cdot 42 + (- y^{2} + 4 a + 42) \cdot 2 b & (distributive multiply) \\ = (−3) a y^{2} + 3 a \cdot 4 a + 3 a \cdot 42 + (- y^{2} + 4 a + 42) \cdot 2 b & (multiply constants) \\ = - 3 a y^{2} + 12 a a + 3 a \cdot 42 + (- y^{2} + 4 a + 42) \cdot 2 b & (multiply constants) \\ = - 3 a y^{2} + 12 a^{2} + 3 a \cdot 42 + (- y^{2} + 4 a + 42) \cdot 2 b & (multiply terms to exponential form) \\ = - 3 a y^{2} + 12 a^{2} + 126 a + (- y^{2} + 4 a + 42) \cdot 2 b & (multiply constants) \\ = - 3 a y^{2} + 12 a^{2} + 126 a + 2 b (- y^{2}) + 2 b \cdot 4 a + 2 b \cdot 42 & (distributive multiply) \\ = - 3 a y^{2} + 12 a^{2} + 126 a + (−2) b y^{2} + 2 b \cdot 4 a + 2 b \cdot 42 & (multiply constants) \\ = - 3 a y^{2} + 12 a^{2} + 126 a - 2 b y^{2} + 8 b a + 2 b \cdot 42 & (multiply constants) \\ = - 3 a y^{2} + 12 a^{2} + 126 a - 2 b y^{2} + 8 a b + 84 b & (multiply constants) \\ = - 3 a y^{2} - 2 b y^{2} + 12 a^{2} + 8 a b + 126 a + 84 b & (reorder terms) \end{array}$

(82)

$ExpandSteps (3(a^2-1)/(a+1), implicitmultiply) &semi;$

$\begin{array}{c} \frac{3 (a^{2} - 1)}{a + 1} \\ = \frac{3 a^{2} + 3 (−1)}{a + 1} & (distributive multiply) \\ = \frac{3 a^{2} - 3}{a + 1} & (multiply constants) \\ = \frac{(a + 1) (3 a - 3)}{a + 1} & (factor) \\ = 3 a - 3 & (divide) \end{array}$

(83)

$ExpandSteps (a-b-c^2-1/2-a/2-(a^2-4)^2) &semi;$

$\begin{array}{c} a - b - c^{2} - \frac{1}{2} - \frac{a}{2} - {(a^{2} - 4)}^{2} \\ = a - b - c^{2} - \frac{1}{2} - \frac{a}{2} - (a^{2} - 4) (a^{2} - 4) & (rewrite exponentiation as multiplication) \\ = a - b - c^{2} - \frac{1}{2} - \frac{a}{2} - (a^{2} (a^{2} - 4) + (−4) (a^{2} - 4)) & (distributive multiply) \\ = a - b - c^{2} - \frac{1}{2} - \frac{a}{2} - (a^{2} a^{2} + a^{2} (−4) + (−4) (a^{2} - 4)) & (distributive multiply) \\ = a - b - c^{2} - \frac{1}{2} - \frac{a}{2} - (a^{4} + a^{2} (−4) + (−4) (a^{2} - 4)) & (add exponents with common base) \\ = a - b - c^{2} - \frac{1}{2} - \frac{a}{2} - (a^{4} - 4 a^{2} + (−4) a^{2} + (−4) (−4)) & (distributive multiply) \\ = a - b - c^{2} - \frac{1}{2} - \frac{a}{2} - (a^{4} - 4 a^{2} - 4 a^{2} + 16) & (multiply constants) \\ = a - b - c^{2} - \frac{1}{2} - \frac{a}{2} - a^{4} + 8 a^{2} - 16 & (add terms) \\ = - \frac{33}{2} + \frac{a}{2} - b - c^{2} - a^{4} + 8 a^{2} & (add terms) \end{array}$

(84)

$ExpandSteps (x^2 y / (xy), implicitmultiply) &semi;$

$\begin{array}{c} \frac{x^{2} y}{x y} \\ = \frac{x^{2}}{x} & (divide out common terms) \\ = x & (divide) \end{array}$

(85)

$ExpandSteps (2x^2*y/(4xy), implicitmultiply) &semi;$

$\begin{array}{c} \frac{2 x^{2} y}{4 x y} \\ = \frac{2 x y}{4 y} & (divide out common terms) \\ = \frac{2 x}{4} & (divide out common terms) \\ = \frac{x}{2} & (reduce fraction by gcd) \end{array}$

(86)

$ExpandSteps (`%*` (`%*` (3, a), 42)) &semi;$

$\begin{array}{c} 3 a \cdot 42 \\ = 126 a & (multiply constants) \end{array}$

(87)

$ExpandSteps (3a * (4a-y+42), implicitmultiply) &semi;$

(88)

$ExpandSteps ((3*a+2*b)^3) &semi;$

$\begin{array}{c} {(3 a + 2 b)}^{3} \\ = (3 a + 2 b) (3 a + 2 b) (3 a + 2 b) & (rewrite exponentiation as multiplication) \\ = (3 a (3 a + 2 b) + 2 b (3 a + 2 b)) (3 a + 2 b) & (distributive multiply) \\ = (3 a \cdot 3 a + 3 a \cdot 2 b + 2 b (3 a + 2 b)) (3 a + 2 b) & (distributive multiply) \\ = (9 a a + 3 a \cdot 2 b + 2 b (3 a + 2 b)) (3 a + 2 b) & (multiply constants) \\ = (9 a^{2} + 3 a \cdot 2 b + 2 b (3 a + 2 b)) (3 a + 2 b) & (multiply terms to exponential form) \\ = (9 a^{2} + 6 a b + 2 b (3 a + 2 b)) (3 a + 2 b) & (multiply constants) \\ = (9 a^{2} + 6 a b + 2 b \cdot 3 a + 2 b \cdot 2 b) (3 a + 2 b) & (distributive multiply) \\ = (9 a^{2} + 6 a b + 6 b a + 2 b \cdot 2 b) (3 a + 2 b) & (multiply constants) \\ = (9 a^{2} + 6 a b + 6 a b + 4 b b) (3 a + 2 b) & (multiply constants) \\ = (9 a^{2} + 6 a b + 6 a b + 4 b^{2}) (3 a + 2 b) & (multiply terms to exponential form) \\ = (9 a^{2} + 12 a b + 4 b^{2}) (3 a + 2 b) & (add terms) \\ = (9 a^{2} + 12 a b + 4 b^{2}) \cdot 3 a + (9 a^{2} + 12 a b + 4 b^{2}) \cdot 2 b & (distributive multiply) \\ = 3 a \cdot 9 a^{2} + 3 a \cdot 12 a b + 3 a \cdot 4 b^{2} + (9 a^{2} + 12 a b + 4 b^{2}) \cdot 2 b & (distributive multiply) \\ = 27 a a^{2} + 3 a \cdot 12 a b + 3 a \cdot 4 b^{2} + (9 a^{2} + 12 a b + 4 b^{2}) \cdot 2 b & (multiply constants) \\ = 27 a^{3} + 3 a \cdot 12 a b + 3 a \cdot 4 b^{2} + (9 a^{2} + 12 a b + 4 b^{2}) \cdot 2 b & (add exponents with common base) \\ = 27 a^{3} + 36 a a b + 3 a \cdot 4 b^{2} + (9 a^{2} + 12 a b + 4 b^{2}) \cdot 2 b & (multiply constants) \\ = 27 a^{3} + 36 a^{2} b + 3 a \cdot 4 b^{2} + (9 a^{2} + 12 a b + 4 b^{2}) \cdot 2 b & (multiply terms to exponential form) \\ = 27 a^{3} + 36 a^{2} b + 12 a b^{2} + (9 a^{2} + 12 a b + 4 b^{2}) \cdot 2 b & (multiply constants) \\ = 27 a^{3} + 36 a^{2} b + 12 a b^{2} + 2 b \cdot 9 a^{2} + 2 b \cdot 12 a b + 2 b \cdot 4 b^{2} & (distributive multiply) \\ = 27 a^{3} + 36 a^{2} b + 12 a b^{2} + 18 b a^{2} + 2 b \cdot 12 a b + 2 b \cdot 4 b^{2} & (multiply constants) \\ = 27 a^{3} + 36 a^{2} b + 12 a b^{2} + 18 a^{2} b + 24 b a b + 2 b \cdot 4 b^{2} & (multiply constants) \\ = 27 a^{3} + 36 a^{2} b + 12 a b^{2} + 18 a^{2} b + 24 b^{2} a + 2 b \cdot 4 b^{2} & (multiply terms to exponential form) \\ = 27 a^{3} + 36 a^{2} b + 12 a b^{2} + 18 a^{2} b + 24 a b^{2} + 8 b b^{2} & (multiply constants) \\ = 27 a^{3} + 36 a^{2} b + 12 a b^{2} + 18 a^{2} b + 24 a b^{2} + 8 b^{3} & (add exponents with common base) \\ = 27 a^{3} + 54 a^{2} b + 36 a b^{2} + 8 b^{3} & (add terms) \end{array}$

(89)

$ExpandSteps ((3*a+2*b)*(4*a-y+42)) &semi;$

(90)

$ExpandSteps ((3*a)*(4*a-y+42)) &semi;$

(91)

$ExpandSteps (3(x-2), implicitmultiply) &semi;$

$\begin{array}{c} 3 (x - 2) \\ = 3 x + 3 (−2) & (distributive multiply) \\ = 3 x - 6 & (multiply constants) \end{array}$

(92)

$ExpandSteps (-(1+y)) &semi;$

$\begin{array}{c} - (1 + y) \\ = −1 - y & (distribute negation) \end{array}$

(93)

$ExpandSteps (-(1+y+y^2)) &semi;$

$\begin{array}{c} - (1 + y + y^{2}) \\ = - y^{2} - y - 1 & (distribute negation) \end{array}$

(94)

$ExpandSteps (y⋅((3+1)/(2*y) - 3*x/y)) &semi;$

$\begin{array}{c} y (\frac{3 + 1}{2 y} - \frac{3 x}{y}) \\ = y (\frac{4}{2 y} - \frac{3 x}{y}) & (add terms) \\ = y (\frac{2}{y} - \frac{3 x}{y}) & (reduce fraction by gcd) \\ = y \frac{2}{y} + y (- \frac{3 x}{y}) & (distributive multiply) \\ = \frac{2 y}{y} + y (- \frac{3 x}{y}) & (multiply fraction) \\ = 2 + y (- \frac{3 x}{y}) & (divide) \\ = 2 + \frac{- 3 x y}{y} & (multiply fraction) \\ = 2 - 3 x & (divide) \end{array}$

(95)

$ExpandSteps (-3x -2x, implicitmultiply) &semi;$

$\begin{array}{c} - 3 x - 2 x \\ = - 5 x & (add terms) \end{array}$

(96)

$ExpandSteps (\frac{(7 + 3 \cdot x - y / 4)}{z}) &semi;$

$\begin{array}{c} \frac{7 + 3 x + (- \frac{1}{4}) y}{z} \\ = \frac{7 + 3 x + \frac{- y}{4}}{z} & (multiply fraction) \end{array}$

(97)

In this example, the input is not quoted so some automatic simplifications happen before ExpandSteps sees the input.

$ExpandSteps (\frac{17 \cdot x^{4} \cdot y^{2}}{64 \cdot z^{5}} + \frac{(24 \cdot y \cdot z^{2})}{85 \cdot x^{2}} \cdot \frac{x^{2}}{z} \cdot (\frac{z^{3}}{x^{2}})) &semi;$

$\begin{array}{c} \frac{17}{64} x^{4} y^{2} z^{−5} + \frac{24}{85} y z^{4} x^{−2} \\ = \frac{17 x^{4} y^{2}}{64 z^{5}} + \frac{24}{85} y z^{4} x^{−2} & (multiply fraction) \\ = \frac{17 x^{4} y^{2}}{64 z^{5}} + \frac{24 y z^{4}}{85 x^{2}} & (multiply fraction) \end{array}$

(98)

$ExpandSteps (\frac{17 \cdot x^{4} \cdot y^{2}}{64 \cdot z^{5}} \cdot \frac{(24 \cdot y \cdot z^{2})}{85 \cdot x^{2}} \cdot \frac{x^{2}}{z} \cdot (\frac{z^{3}}{x^{2}})) &semi;$

$\begin{array}{c} \frac{\frac{3}{40} x^{2} y^{3}}{z} \\ = \frac{\frac{3 x^{2} y^{3}}{40}}{z} & (multiply fraction) \\ = \frac{3 x^{2} y^{3}}{40 z} & (reduce fraction by gcd) \end{array}$

(99)

$ExpandSteps ("((17*x^4*y^2)/(64*z^5)) * ((24 \cdot y \cdot z &Hat; 2) / (85 \cdot x &Hat; 2)) &ast; (x &Hat; 2 / z) &ast; (z &Hat; 3 / x &Hat; 2) ");$

$\begin{array}{c} \frac{17 x^{4} y^{2}}{64 z^{5}} \frac{24 y z^{2}}{85 x^{2}} \frac{x^{2}}{z} \frac{z^{3}}{x^{2}} \\ = \frac{408 x^{6} y^{3} z^{5}}{5440 z^{6} x^{4}} & (multiply fraction) \\ = \frac{408 x^{6} y^{3}}{5440 z x^{4}} & (divide out common terms) \\ = \frac{408 x^{2} y^{3}}{5440 z} & (divide out common terms) \\ = \frac{3 x^{2} y^{3}}{40 z} & (reduce fraction by gcd) \end{array}$

(100)

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