Algebra Examples

${(a^{- \frac{1}{4}} b)}^{- 4} {(b^{2} c)}^{- \frac{1}{4}}$

Step 1

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

${(\frac{1}{a^{\frac{1}{4}}} b)}^{- 4} {(b^{2} c)}^{- \frac{1}{4}}$

Step 2

Combine $\frac{1}{a^{\frac{1}{4}}}$ and $b$ .

${(\frac{b}{a^{\frac{1}{4}}})}^{- 4} {(b^{2} c)}^{- \frac{1}{4}}$

Step 3

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

${(\frac{a^{\frac{1}{4}}}{b})}^{4} {(b^{2} c)}^{- \frac{1}{4}}$

Step 4

Apply the product rule to $\frac{a^{\frac{1}{4}}}{b}$ .

$\frac{{(a^{\frac{1}{4}})}^{4}}{b^{4}} {(b^{2} c)}^{- \frac{1}{4}}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

$\frac{a^{\frac{1}{4} \cdot 4}}{b^{4}} {(b^{2} c)}^{- \frac{1}{4}}$

Step 5.1.2

Cancel the common factor of $4$ .

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Step 5.1.2.1

Cancel the common factor.

$\frac{a^{\frac{1}{4} \cdot 4}}{b^{4}} {(b^{2} c)}^{- \frac{1}{4}}$

Step 5.1.2.2

Rewrite the expression.

$\frac{a^{1}}{b^{4}} {(b^{2} c)}^{- \frac{1}{4}}$

Step 5.2

Simplify.

$\frac{a}{b^{4}} {(b^{2} c)}^{- \frac{1}{4}}$

Step 6

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

$\frac{a}{b^{4}} \cdot \frac{1}{{(b^{2} c)}^{\frac{1}{4}}}$

Step 7

Combine fractions.

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Step 7.1

Combine.

$\frac{a \cdot 1}{b^{4} {(b^{2} c)}^{\frac{1}{4}}}$

Step 7.2

Multiply $a$ by $1$ .

$\frac{a}{b^{4} {(b^{2} c)}^{\frac{1}{4}}}$

Step 8

Simplify the denominator.

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Step 8.1

Apply the product rule to $b^{2} c$ .

$\frac{a}{b^{4} ({(b^{2})}^{\frac{1}{4}} c^{\frac{1}{4}})}$

Step 8.2

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

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Step 8.2.1

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

$\frac{a}{b^{4} (b^{2 (\frac{1}{4})} c^{\frac{1}{4}})}$

Step 8.2.2

Cancel the common factor of $2$ .

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Step 8.2.2.1

Factor $2$ out of $4$ .

$\frac{a}{b^{4} (b^{2 \frac{1}{2 (2)}} c^{\frac{1}{4}})}$

Step 8.2.2.2

Cancel the common factor.

$\frac{a}{b^{4} (b^{2 \frac{1}{2 \cdot 2}} c^{\frac{1}{4}})}$

Step 8.2.2.3

Rewrite the expression.

$\frac{a}{b^{4} (b^{\frac{1}{2}} c^{\frac{1}{4}})}$

$\frac{a}{b^{4} b^{\frac{1}{2}} c^{\frac{1}{4}}}$

Step 8.3

Multiply $b^{4}$ by $b^{\frac{1}{2}}$ by adding the exponents.

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Step 8.3.1

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

$\frac{a}{b^{4 + \frac{1}{2}} c^{\frac{1}{4}}}$

Step 8.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{a}{b^{4 \cdot \frac{2}{2} + \frac{1}{2}} c^{\frac{1}{4}}}$

Step 8.3.3

Combine $4$ and $\frac{2}{2}$ .

$\frac{a}{b^{\frac{4 \cdot 2}{2} + \frac{1}{2}} c^{\frac{1}{4}}}$

Step 8.3.4

Combine the numerators over the common denominator.

$\frac{a}{b^{\frac{4 \cdot 2 + 1}{2}} c^{\frac{1}{4}}}$

Step 8.3.5

Simplify the numerator.

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Step 8.3.5.1

Multiply $4$ by $2$ .

$\frac{a}{b^{\frac{8 + 1}{2}} c^{\frac{1}{4}}}$

Step 8.3.5.2

Add $8$ and $1$ .

$\frac{a}{b^{\frac{9}{2}} c^{\frac{1}{4}}}$