Algebra Examples

${(\frac{z^{- \frac{4}{3}}}{12 x^{3} y^{\frac{1}{4}}})}^{- \frac{1}{2}}$

Step 1

Move $z^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{12 x^{3} y^{\frac{1}{4}} z^{\frac{4}{3}}})}^{- \frac{1}{2}}$

Step 2

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

${(12 x^{3} y^{\frac{1}{4}} z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 3

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

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Step 3.1

Apply the product rule to $12 x^{3} y^{\frac{1}{4}} z^{\frac{4}{3}}$ .

${(12 x^{3} y^{\frac{1}{4}})}^{\frac{1}{2}} {(z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 3.2

Apply the product rule to $12 x^{3} y^{\frac{1}{4}}$ .

${(12 x^{3})}^{\frac{1}{2}} {(y^{\frac{1}{4}})}^{\frac{1}{2}} {(z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 3.3

Apply the product rule to $12 x^{3}$ .

$12^{\frac{1}{2}} {(x^{3})}^{\frac{1}{2}} {(y^{\frac{1}{4}})}^{\frac{1}{2}} {(z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 4

Multiply the exponents in ${(x^{3})}^{\frac{1}{2}}$ .

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Step 4.1

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

$12^{\frac{1}{2}} x^{3 (\frac{1}{2})} {(y^{\frac{1}{4}})}^{\frac{1}{2}} {(z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 4.2

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

$12^{\frac{1}{2}} x^{\frac{3}{2}} {(y^{\frac{1}{4}})}^{\frac{1}{2}} {(z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 5

Multiply the exponents in ${(y^{\frac{1}{4}})}^{\frac{1}{2}}$ .

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Step 5.1

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

$12^{\frac{1}{2}} x^{\frac{3}{2}} y^{\frac{1}{4} \cdot \frac{1}{2}} {(z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 5.2

Multiply $\frac{1}{4} \cdot \frac{1}{2}$ .

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Step 5.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$12^{\frac{1}{2}} x^{\frac{3}{2}} y^{\frac{1}{4 \cdot 2}} {(z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 5.2.2

Multiply $4$ by $2$ .

$12^{\frac{1}{2}} x^{\frac{3}{2}} y^{\frac{1}{8}} {(z^{\frac{4}{3}})}^{\frac{1}{2}}$

Step 6

Multiply the exponents in ${(z^{\frac{4}{3}})}^{\frac{1}{2}}$ .

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Step 6.1

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

$12^{\frac{1}{2}} x^{\frac{3}{2}} y^{\frac{1}{8}} z^{\frac{4}{3} \cdot \frac{1}{2}}$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $4$ .

$12^{\frac{1}{2}} x^{\frac{3}{2}} y^{\frac{1}{8}} z^{\frac{2 (2)}{3} \cdot \frac{1}{2}}$

Step 6.2.2

Cancel the common factor.

$12^{\frac{1}{2}} x^{\frac{3}{2}} y^{\frac{1}{8}} z^{\frac{2 \cdot 2}{3} \cdot \frac{1}{2}}$

Step 6.2.3

Rewrite the expression.

$12^{\frac{1}{2}} x^{\frac{3}{2}} y^{\frac{1}{8}} z^{\frac{2}{3}}$