Algebra Examples

${(\frac{1}{4} x^{3} y^{- 2})}^{- 2} {(- 4 x^{- 1} y)}^{- 3}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Combine $x^{3}$ and $\frac{1}{y^{2}}$ .

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

Step 3

Combine.

${(\frac{1 x^{3}}{4 y^{2}})}^{- 2} {(- 4 x^{- 1} y)}^{- 3}$

Step 4

Multiply $x^{3}$ by $1$ .

${(\frac{x^{3}}{4 y^{2}})}^{- 2} {(- 4 x^{- 1} y)}^{- 3}$

Step 5

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{4 y^{2}}{x^{3}})}^{2} {(- 4 x^{- 1} y)}^{- 3}$

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.1

Apply the product rule to $\frac{4 y^{2}}{x^{3}}$ .

$\frac{{(4 y^{2})}^{2}}{{(x^{3})}^{2}} {(- 4 x^{- 1} y)}^{- 3}$

Step 6.2

Apply the product rule to $4 y^{2}$ .

$\frac{4^{2} {(y^{2})}^{2}}{{(x^{3})}^{2}} {(- 4 x^{- 1} y)}^{- 3}$

Step 7

Simplify the numerator.

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Step 7.1

Raise $4$ to the power of $2$ .

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

Step 7.2

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 7.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.2.2

Multiply $2$ by $2$ .

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

Step 8

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 8.2

Multiply $3$ by $2$ .

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

Step 9

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10

Combine $- 4$ and $\frac{1}{x}$ .

$\frac{16 y^{4}}{x^{6}} {(\frac{- 4}{x} y)}^{- 3}$

Step 11

Move the negative in front of the fraction.

$\frac{16 y^{4}}{x^{6}} {(- \frac{4}{x} y)}^{- 3}$

Step 12

Combine $y$ and $\frac{4}{x}$ .

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

Step 13

Move $4$ to the left of $y$ .

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

Step 14

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{16 y^{4}}{x^{6}} {(- \frac{x}{4 y})}^{3}$

Step 15

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 15.1

Apply the product rule to $- \frac{x}{4 y}$ .

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

Step 15.2

Apply the product rule to $\frac{x}{4 y}$ .

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

Step 15.3

Apply the product rule to $4 y$ .

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

Step 16

Simplify the expression.

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Step 16.1

Rewrite using the commutative property of multiplication.

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

Step 16.2

Raise $- 1$ to the power of $3$ .

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

Step 17

Cancel the common factor of $y^{3}$ .

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Step 17.1

Move the leading negative in $- \frac{16 y^{4}}{x^{6}}$ into the numerator.

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

Step 17.2

Factor $y^{3}$ out of $- 16 y^{4}$ .

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

Step 17.3

Factor $y^{3}$ out of $4^{3} y^{3}$ .

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

Step 17.4

Cancel the common factor.

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

Step 17.5

Rewrite the expression.

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

Step 18

Cancel the common factor of $x^{3}$ .

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Step 18.1

Factor $x^{3}$ out of $x^{6}$ .

$\frac{- 16 y}{x^{3} x^{3}} \cdot \frac{x^{3}}{4^{3}}$

Step 18.2

Cancel the common factor.

$\frac{- 16 y}{x^{3} x^{3}} \cdot \frac{x^{3}}{4^{3}}$

Step 18.3

Rewrite the expression.

$\frac{- 16 y}{x^{3}} \cdot \frac{1}{4^{3}}$

Step 19

Multiply $\frac{- 16 y}{x^{3}}$ by $\frac{1}{4^{3}}$ .

$\frac{- 16 y}{x^{3} \cdot 4^{3}}$

Step 20

Raise $4$ to the power of $3$ .

$\frac{- 16 y}{x^{3} \cdot 64}$

Step 21

Cancel the common factor of $- 16$ and $64$ .

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Step 21.1

Factor $16$ out of $- 16 y$ .

$\frac{16 (- y)}{x^{3} \cdot 64}$

Step 21.2

Cancel the common factors.

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Step 21.2.1

Factor $16$ out of $x^{3} \cdot 64$ .

$\frac{16 (- y)}{16 (x^{3} \cdot 4)}$

Step 21.2.2

Cancel the common factor.

$\frac{16 (- y)}{16 (x^{3} \cdot 4)}$

Step 21.2.3

Rewrite the expression.

$\frac{- y}{x^{3} \cdot 4}$

Step 22

Simplify the expression.

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Step 22.1

Move $4$ to the left of $x^{3}$ .

$\frac{- y}{4 \cdot x^{3}}$

Step 22.2

Move the negative in front of the fraction.

$- \frac{y}{4 x^{3}}$