Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

Combine.

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

Step 3.2

Multiply $\frac{1}{y^{2}}$ by $1$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Apply the product rule to $- 2 x y^{3} z^{2}$ .

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

Step 4.2

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

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Step 4.2.1

Apply the product rule to $- 2 x y^{3}$ .

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

Step 4.2.2

Apply the product rule to $- 2 x$ .

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Multiply $3$ by $2$ .

$\frac{\frac{1}{y^{2}}}{4 x^{2} y^{6} {(z^{2})}^{2} z^{2}}$

Step 4.5

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

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Step 4.5.1

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

$\frac{\frac{1}{y^{2}}}{4 x^{2} y^{6} z^{2 \cdot 2} z^{2}}$

Step 4.5.2

Multiply $2$ by $2$ .

$\frac{\frac{1}{y^{2}}}{4 x^{2} y^{6} z^{4} z^{2}}$

Step 4.6

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

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Step 4.6.1

Move $z^{2}$ .

$\frac{\frac{1}{y^{2}}}{4 x^{2} y^{6} (z^{2} z^{4})}$

Step 4.6.2

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

$\frac{\frac{1}{y^{2}}}{4 x^{2} y^{6} z^{2 + 4}}$

Step 4.6.3

Add $2$ and $4$ .

$\frac{\frac{1}{y^{2}}}{4 x^{2} y^{6} z^{6}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{y^{2}} \cdot \frac{1}{4 x^{2} y^{6} z^{6}}$

Step 6

Combine.

$\frac{1 \cdot 1}{y^{2} (4 x^{2} y^{6} z^{6})}$

Step 7

Multiply $y^{2}$ by $y^{6}$ by adding the exponents.

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Step 7.1

Move $y^{6}$ .

$\frac{1 \cdot 1}{y^{6} y^{2} (4 x^{2} z^{6})}$

Step 7.2

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

$\frac{1 \cdot 1}{y^{6 + 2} (4 x^{2} z^{6})}$

Step 7.3

Add $6$ and $2$ .

$\frac{1 \cdot 1}{y^{8} (4 x^{2} z^{6})}$

Step 8

Multiply $1$ by $1$ .

$\frac{1}{y^{8} (4 x^{2} z^{6})}$

Step 9

Move $4$ to the left of $y^{8}$ .

$\frac{1}{4 y^{8} x^{2} z^{6}}$