Pre-Algebra Examples

${(\frac{3 x}{4} + \frac{4 y}{3} - 6)}^{2}$

Step 1

Rewrite ${(\frac{3 x}{4} + \frac{4 y}{3} - 6)}^{2}$ as $(\frac{3 x}{4} + \frac{4 y}{3} - 6) (\frac{3 x}{4} + \frac{4 y}{3} - 6)$ .

$(\frac{3 x}{4} + \frac{4 y}{3} - 6) (\frac{3 x}{4} + \frac{4 y}{3} - 6)$

Step 2

Expand $(\frac{3 x}{4} + \frac{4 y}{3} - 6) (\frac{3 x}{4} + \frac{4 y}{3} - 6)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{3 x}{4} \cdot \frac{3 x}{4} + \frac{3 x}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\frac{3 x}{4} \cdot \frac{3 x}{4}$ .

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Step 3.1.1.1

Multiply $\frac{3 x}{4}$ by $\frac{3 x}{4}$ .

$\frac{3 x (3 x)}{4 \cdot 4} + \frac{3 x}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.1.2

Multiply $3$ by $3$ .

$\frac{9 x \cdot x}{4 \cdot 4} + \frac{3 x}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.1.3

Raise $x$ to the power of $1$ .

$\frac{9 (x^{1} x)}{4 \cdot 4} + \frac{3 x}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.1.4

Raise $x$ to the power of $1$ .

$\frac{9 (x^{1} x^{1})}{4 \cdot 4} + \frac{3 x}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 x^{1 + 1}}{4 \cdot 4} + \frac{3 x}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.1.6

Add $1$ and $1$ .

$\frac{9 x^{2}}{4 \cdot 4} + \frac{3 x}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.1.7

Multiply $4$ by $4$ .

$\frac{9 x^{2}}{16} + \frac{3 x}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.2

Cancel the common factor of $3$ .

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Step 3.1.2.1

Factor $3$ out of $3 x$ .

$\frac{9 x^{2}}{16} + \frac{3 (x)}{4} \cdot \frac{4 y}{3} + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.2.2

Cancel the common factor.

Step 3.1.2.3

Rewrite the expression.

$\frac{9 x^{2}}{16} + \frac{x}{4} (4 y) + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.3

Cancel the common factor of $4$ .

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Step 3.1.3.1

Factor $4$ out of $4 y$ .

$\frac{9 x^{2}}{16} + \frac{x}{4} (4 (y)) + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.3.2

Cancel the common factor.

Step 3.1.3.3

Rewrite the expression.

$\frac{9 x^{2}}{16} + x y + \frac{3 x}{4} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Factor $2$ out of $4$ .

$\frac{9 x^{2}}{16} + x y + \frac{3 x}{2 (2)} \cdot - 6 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.4.2

Factor $2$ out of $- 6$ .

$\frac{9 x^{2}}{16} + x y + \frac{3 x}{2 \cdot 2} \cdot (2 \cdot - 3) + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.4.3

Cancel the common factor.

Step 3.1.4.4

Rewrite the expression.

$\frac{9 x^{2}}{16} + x y + \frac{3 x}{2} \cdot - 3 + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.5

Combine $\frac{3 x}{2}$ and $- 3$ .

$\frac{9 x^{2}}{16} + x y + \frac{3 x \cdot - 3}{2} + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.6

Multiply $- 3$ by $3$ .

$\frac{9 x^{2}}{16} + x y + \frac{- 9 x}{2} + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.7

Move the negative in front of the fraction.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.8

Cancel the common factor of $4$ .

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Step 3.1.8.1

Factor $4$ out of $4 y$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + \frac{4 (y)}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.8.2

Cancel the common factor.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + \frac{4 y}{3} \cdot \frac{3 x}{4} + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.8.3

Rewrite the expression.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + \frac{y}{3} (3 x) + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.9

Cancel the common factor of $3$ .

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Step 3.1.9.1

Factor $3$ out of $3 x$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + \frac{y}{3} (3 (x)) + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.9.2

Cancel the common factor.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + \frac{y}{3} (3 x) + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.9.3

Rewrite the expression.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{4 y}{3} \cdot \frac{4 y}{3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.10

Multiply $\frac{4 y}{3} \cdot \frac{4 y}{3}$ .

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Step 3.1.10.1

Multiply $\frac{4 y}{3}$ by $\frac{4 y}{3}$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{4 y (4 y)}{3 \cdot 3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.10.2

Multiply $4$ by $4$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y \cdot y}{3 \cdot 3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.10.3

Raise $y$ to the power of $1$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 (y^{1} y)}{3 \cdot 3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.10.4

Raise $y$ to the power of $1$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 (y^{1} y^{1})}{3 \cdot 3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.10.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{1 + 1}}{3 \cdot 3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.10.6

Add $1$ and $1$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{3 \cdot 3} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.10.7

Multiply $3$ by $3$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} + \frac{4 y}{3} \cdot - 6 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.11

Cancel the common factor of $3$ .

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Step 3.1.11.1

Factor $3$ out of $- 6$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} + \frac{4 y}{3} \cdot (3 (- 2)) - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.11.2

Cancel the common factor.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} + \frac{4 y}{3} \cdot (3 \cdot - 2) - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.11.3

Rewrite the expression.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} + 4 y \cdot - 2 - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.12

Multiply $- 2$ by $4$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y - 6 \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.13

Cancel the common factor of $2$ .

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Step 3.1.13.1

Factor $2$ out of $- 6$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y + 2 (- 3) \frac{3 x}{4} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.13.2

Factor $2$ out of $4$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y + 2 \cdot - 3 \frac{3 x}{2 \cdot 2} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.13.3

Cancel the common factor.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y + 2 \cdot - 3 \frac{3 x}{2 \cdot 2} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.13.4

Rewrite the expression.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y - 3 \frac{3 x}{2} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.14

Combine $- 3$ and $\frac{3 x}{2}$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y + \frac{- 3 (3 x)}{2} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.15

Multiply $3$ by $- 3$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y + \frac{- 9 x}{2} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.16

Move the negative in front of the fraction.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y - \frac{9 x}{2} - 6 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.17

Cancel the common factor of $3$ .

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Step 3.1.17.1

Factor $3$ out of $- 6$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y - \frac{9 x}{2} + 3 (- 2) \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.17.2

Cancel the common factor.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y - \frac{9 x}{2} + 3 \cdot - 2 \frac{4 y}{3} - 6 \cdot - 6$

Step 3.1.17.3

Rewrite the expression.

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y - \frac{9 x}{2} - 2 (4 y) - 6 \cdot - 6$

Step 3.1.18

Multiply $4$ by $- 2$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y - \frac{9 x}{2} - 8 y - 6 \cdot - 6$

Step 3.1.19

Multiply $- 6$ by $- 6$ .

$\frac{9 x^{2}}{16} + x y - \frac{9 x}{2} + y x + \frac{16 y^{2}}{9} - 8 y - \frac{9 x}{2} - 8 y + 36$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

$x y + y x - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} + \frac{- 9 x - 9 x}{2} + \frac{16 y^{2}}{9}$

Step 3.2.2

Subtract $9 x$ from $- 9 x$ .

$x y + y x - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} + \frac{- 18 x}{2} + \frac{16 y^{2}}{9}$

Step 3.2.3

Cancel the common factor of $- 18$ and $2$ .

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Step 3.2.3.1

Factor $2$ out of $- 18 x$ .

$x y + y x - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} + \frac{2 (- 9 x)}{2} + \frac{16 y^{2}}{9}$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor $2$ out of $2$ .

$x y + y x - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} + \frac{2 (- 9 x)}{2 (1)} + \frac{16 y^{2}}{9}$

Step 3.2.3.2.2

Cancel the common factor.

$x y + y x - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} + \frac{2 (- 9 x)}{2 \cdot 1} + \frac{16 y^{2}}{9}$

Step 3.2.3.2.3

Rewrite the expression.

$x y + y x - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} + \frac{- 9 x}{1} + \frac{16 y^{2}}{9}$

Step 3.2.3.2.4

Divide $- 9 x$ by $1$ .

$x y + y x - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} - 9 x + \frac{16 y^{2}}{9}$

Step 4

Add $x y$ and $y x$ .

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Step 4.1

Reorder $y$ and $x$ .

$x y + x y - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} - 9 x + \frac{16 y^{2}}{9}$

Step 4.2

Add $x y$ and $x y$ .

$2 x y - 8 y - 8 y + 36 + \frac{9 x^{2}}{16} - 9 x + \frac{16 y^{2}}{9}$

Step 5

Subtract $8 y$ from $- 8 y$ .

$2 x y - 16 y + 36 + \frac{9 x^{2}}{16} - 9 x + \frac{16 y^{2}}{9}$

Step 6

Simplify the expression.

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Step 6.1

Move $36$ .

$2 x y - 16 y + \frac{9 x^{2}}{16} - 9 x + \frac{16 y^{2}}{9} + 36$

Step 6.2

Move $- 16 y$ .

$2 x y + \frac{9 x^{2}}{16} - 9 x + \frac{16 y^{2}}{9} - 16 y + 36$

Step 6.3

Move $- 9 x$ .

$2 x y + \frac{9 x^{2}}{16} + \frac{16 y^{2}}{9} - 9 x - 16 y + 36$

Step 6.4

Reorder $2 x y$ and $\frac{9 x^{2}}{16}$ .

$\frac{9 x^{2}}{16} + 2 x y + \frac{16 y^{2}}{9} - 9 x - 16 y + 36$