Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

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

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

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 $\frac{3 u z^{2}}{x^{2}}$ .

$\frac{x^{3}}{z} \cdot \frac{{(3 u z^{2})}^{2}}{{(x^{2})}^{2}}$

Step 4.2

Apply the product rule to $3 u z^{2}$ .

$\frac{x^{3}}{z} \cdot \frac{{(3 u)}^{2} {(z^{2})}^{2}}{{(x^{2})}^{2}}$

Step 4.3

Apply the product rule to $3 u$ .

$\frac{x^{3}}{z} \cdot \frac{3^{2} u^{2} {(z^{2})}^{2}}{{(x^{2})}^{2}}$

Step 5

Combine.

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

Step 6

Simplify the numerator.

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Step 6.1

Raise $3$ to the power of $2$ .

$\frac{x^{3} \cdot 9 u^{2} {(z^{2})}^{2}}{z {(x^{2})}^{2}}$

Step 6.2

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

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Step 6.2.1

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

$\frac{x^{3} \cdot 9 u^{2} z^{2 \cdot 2}}{z {(x^{2})}^{2}}$

Step 6.2.2

Multiply $2$ by $2$ .

$\frac{x^{3} \cdot 9 u^{2} z^{4}}{z {(x^{2})}^{2}}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

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

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Step 7.1.1

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

$\frac{x^{3} \cdot 9 u^{2} z^{4}}{z x^{2 \cdot 2}}$

Step 7.1.2

Multiply $2$ by $2$ .

$\frac{x^{3} \cdot 9 u^{2} z^{4}}{z x^{4}}$

Step 7.2

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 7.2.1

Factor $x^{3}$ out of $x^{3} \cdot 9 u^{2} z^{4}$ .

$\frac{x^{3} (9 u^{2} z^{4})}{z x^{4}}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $x^{3}$ out of $z x^{4}$ .

$\frac{x^{3} (9 u^{2} z^{4})}{x^{3} (z x)}$

Step 7.2.2.2

Cancel the common factor.

$\frac{x^{3} (9 u^{2} z^{4})}{x^{3} (z x)}$

Step 7.2.2.3

Rewrite the expression.

$\frac{9 u^{2} z^{4}}{z x}$

Step 7.3

Cancel the common factor of $z^{4}$ and $z$ .

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Step 7.3.1

Factor $z$ out of $9 u^{2} z^{4}$ .

$\frac{z (9 u^{2} z^{3})}{z x}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $z$ out of $z x$ .

$\frac{z (9 u^{2} z^{3})}{z (x)}$

Step 7.3.2.2

Cancel the common factor.

$\frac{z (9 u^{2} z^{3})}{z x}$

Step 7.3.2.3

Rewrite the expression.

$\frac{9 u^{2} z^{3}}{x}$