Algebra Examples

$\frac{{(- {(14 x)}^{0} y^{- 7} z)}^{- 2}}{{(y^{- 3} z)}^{- 2}}$

Step 1

Simplify the expression.

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Step 1.1

Move ${(- {(14 x)}^{0} y^{- 7} z)}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(y^{- 3} z)}^{- 2} {(- {(14 x)}^{0} y^{- 7} z)}^{2}}$

Step 1.2

Move ${(y^{- 3} z)}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(y^{- 3} z)}^{2}}{{(- 1 y^{- 7} z)}^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(y^{- 3})}^{2} z^{2}}{{(- 1 y^{- 7} z)}^{2}}$

Step 2.2

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

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Step 2.2.1

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

$\frac{y^{- 3 \cdot 2} z^{2}}{{(- 1 y^{- 7} z)}^{2}}$

Step 2.2.2

Multiply $- 3$ by $2$ .

$\frac{y^{- 6} z^{2}}{{(- 1 y^{- 7} z)}^{2}}$

Step 2.3

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

$\frac{\frac{1}{y^{6}} z^{2}}{{(- 1 y^{- 7} z)}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $- 1 y^{- 7} z$ .

$\frac{\frac{1}{y^{6}} z^{2}}{{(- 1 y^{- 7})}^{2} z^{2}}$

Step 3.2

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

$\frac{\frac{1}{y^{6}} z^{2}}{{(- 1 \frac{1}{y^{7}})}^{2} z^{2}}$

Step 3.3

Rewrite $- 1 \frac{1}{y^{7}}$ as $- \frac{1}{y^{7}}$ .

$\frac{\frac{1}{y^{6}} z^{2}}{{(- \frac{1}{y^{7}})}^{2} z^{2}}$

Step 3.4

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

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Step 3.4.1

Apply the product rule to $- \frac{1}{y^{7}}$ .

$\frac{\frac{1}{y^{6}} z^{2}}{{(- 1)}^{2} {(\frac{1}{y^{7}})}^{2} z^{2}}$

Step 3.4.2

Apply the product rule to $\frac{1}{y^{7}}$ .

$\frac{\frac{1}{y^{6}} z^{2}}{{(- 1)}^{2} \frac{1^{2}}{{(y^{7})}^{2}} z^{2}}$

Step 3.5

Raise $- 1$ to the power of $2$ .

$\frac{\frac{1}{y^{6}} z^{2}}{1 \frac{1^{2}}{{(y^{7})}^{2}} z^{2}}$

Step 3.6

Multiply $\frac{1^{2}}{{(y^{7})}^{2}}$ by $1$ .

$\frac{\frac{1}{y^{6}} z^{2}}{\frac{1^{2}}{{(y^{7})}^{2}} z^{2}}$

Step 3.7

One to any power is one.

$\frac{\frac{1}{y^{6}} z^{2}}{\frac{1}{{(y^{7})}^{2}} z^{2}}$

Step 3.8

Multiply the exponents in ${(y^{7})}^{2}$ .

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Step 3.8.1

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

$\frac{\frac{1}{y^{6}} z^{2}}{\frac{1}{y^{7 \cdot 2}} z^{2}}$

Step 3.8.2

Multiply $7$ by $2$ .

$\frac{\frac{1}{y^{6}} z^{2}}{\frac{1}{y^{14}} z^{2}}$

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{y^{6}}$ and $z^{2}$ .

$\frac{\frac{z^{2}}{y^{6}}}{\frac{1}{y^{14}} z^{2}}$

Step 4.2

Combine $\frac{1}{y^{14}}$ and $z^{2}$ .

$\frac{\frac{z^{2}}{y^{6}}}{\frac{z^{2}}{y^{14}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{z^{2}}{y^{6}} \cdot \frac{y^{14}}{z^{2}}$

Step 6

Combine.

$\frac{z^{2} y^{14}}{y^{6} z^{2}}$

Step 7

Cancel the common factor of $z^{2}$ .

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Step 7.1

Cancel the common factor.

$\frac{z^{2} y^{14}}{y^{6} z^{2}}$

Step 7.2

Rewrite the expression.

$\frac{y^{14}}{y^{6}}$

Step 8

Cancel the common factor of $y^{14}$ and $y^{6}$ .

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Step 8.1

Factor $y^{6}$ out of $y^{14}$ .

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

Step 8.2

Cancel the common factors.

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Step 8.2.1

Multiply by $1$ .

$\frac{y^{6} y^{8}}{y^{6} \cdot 1}$

Step 8.2.2

Cancel the common factor.

$\frac{y^{6} y^{8}}{y^{6} \cdot 1}$

Step 8.2.3

Rewrite the expression.

$\frac{y^{8}}{1}$

Step 8.2.4

Divide $y^{8}$ by $1$ .

$y^{8}$