Algebra Examples

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

Step 1

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

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

Step 2

Multiply $4 x^{2} \frac{1}{y^{2}}$ .

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Step 2.1

Combine $4$ and $\frac{1}{y^{2}}$ .

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

Step 2.2

Combine $x^{2}$ and $\frac{4}{y^{2}}$ .

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

Step 3

Move $4$ to the left of $x^{2}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $2$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Multiply $2$ by $2$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

$\frac{16 x^{4}}{y^{4}} \cdot \frac{x^{3}}{2 y^{2}}$

Step 11

Combine.

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

Step 12

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 12.1

Move $x^{3}$ .

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

Step 12.2

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

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

Step 12.3

Add $3$ and $4$ .

$\frac{16 x^{7}}{y^{4} (2 y^{2})}$

Step 13

Multiply $y^{4}$ by $y^{2}$ by adding the exponents.

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Step 13.1

Move $y^{2}$ .

$\frac{16 x^{7}}{y^{2} y^{4} \cdot 2}$

Step 13.2

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

$\frac{16 x^{7}}{y^{2 + 4} \cdot 2}$

Step 13.3

Add $2$ and $4$ .

$\frac{16 x^{7}}{y^{6} \cdot 2}$

Step 14

Cancel the common factor of $16$ and $2$ .

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Step 14.1

Factor $2$ out of $16 x^{7}$ .

$\frac{2 (8 x^{7})}{y^{6} \cdot 2}$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $2$ out of $y^{6} \cdot 2$ .

$\frac{2 (8 x^{7})}{2 \cdot y^{6}}$

Step 14.2.2

Cancel the common factor.

$\frac{2 (8 x^{7})}{2 \cdot y^{6}}$

Step 14.2.3

Rewrite the expression.

$\frac{8 x^{7}}{y^{6}}$