Algebra Examples

$\frac{{(3 d^{5} x^{- 2})}^{- 4}}{4 d^{- 6} x^{10}}$

Step 1

Move ${(3 d^{5} x^{- 2})}^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{4 d^{- 6} x^{10} {(3 d^{5} x^{- 2})}^{4}}$

Step 2

Move $d^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{d^{6}}{4 x^{10} {(3 d^{5} x^{- 2})}^{4}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $3 d^{5} x^{- 2}$ .

$\frac{d^{6}}{4 x^{10} {(3 d^{5})}^{4} {(x^{- 2})}^{4}}$

Step 3.2

Apply the product rule to $3 d^{5}$ .

$\frac{d^{6}}{4 x^{10} (3^{4} {(d^{5})}^{4}) {(x^{- 2})}^{4}}$

Step 3.3

Raise $3$ to the power of $4$ .

$\frac{d^{6}}{4 x^{10} (81 {(d^{5})}^{4}) {(x^{- 2})}^{4}}$

Step 3.4

Multiply the exponents in ${(d^{5})}^{4}$ .

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Step 3.4.1

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

$\frac{d^{6}}{4 x^{10} (81 d^{5 \cdot 4}) {(x^{- 2})}^{4}}$

Step 3.4.2

Multiply $5$ by $4$ .

$\frac{d^{6}}{4 x^{10} (81 d^{20}) {(x^{- 2})}^{4}}$

Step 3.5

Multiply the exponents in ${(x^{- 2})}^{4}$ .

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Step 3.5.1

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

$\frac{d^{6}}{4 x^{10} (81 d^{20}) x^{- 2 \cdot 4}}$

Step 3.5.2

Multiply $- 2$ by $4$ .

$\frac{d^{6}}{4 x^{10} (81 d^{20}) x^{- 8}}$

Step 3.6

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

$\frac{d^{6}}{4 x^{10} \cdot 81 d^{20} \frac{1}{x^{8}}}$

Step 3.7

Combine exponents.

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Step 3.7.1

Multiply $81$ by $4$ .

$\frac{d^{6}}{324 x^{10} d^{20} \frac{1}{x^{8}}}$

Step 3.7.2

Combine $324$ and $\frac{1}{x^{8}}$ .

$\frac{d^{6}}{x^{10} d^{20} \frac{324}{x^{8}}}$

Step 3.7.3

Combine $x^{10}$ and $\frac{324}{x^{8}}$ .

$\frac{d^{6}}{d^{20} \frac{x^{10} \cdot 324}{x^{8}}}$

Step 3.7.4

Combine $d^{20}$ and $\frac{x^{10} \cdot 324}{x^{8}}$ .

$\frac{d^{6}}{\frac{d^{20} (x^{10} \cdot 324)}{x^{8}}}$

Step 3.8

Remove unnecessary parentheses.

$\frac{d^{6}}{\frac{d^{20} x^{10} \cdot 324}{x^{8}}}$

Step 3.9

Reduce the expression by cancelling the common factors.

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Step 3.9.1

Reduce the expression $\frac{d^{20} x^{10} \cdot 324}{x^{8}}$ by cancelling the common factors.

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Step 3.9.1.1

Factor $x^{8}$ out of $d^{20} x^{10} \cdot 324$ .

$\frac{d^{6}}{\frac{x^{8} (d^{20} x^{2} \cdot 324)}{x^{8}}}$

Step 3.9.1.2

Multiply by $1$ .

$\frac{d^{6}}{\frac{x^{8} (d^{20} x^{2} \cdot 324)}{x^{8} \cdot 1}}$

Step 3.9.1.3

Cancel the common factor.

$\frac{d^{6}}{\frac{x^{8} (d^{20} x^{2} \cdot 324)}{x^{8} \cdot 1}}$

Step 3.9.1.4

Rewrite the expression.

$\frac{d^{6}}{\frac{d^{20} x^{2} \cdot 324}{1}}$

Step 3.9.2

Divide $d^{20} x^{2} \cdot 324$ by $1$ .

$\frac{d^{6}}{d^{20} x^{2} \cdot 324}$

Step 4

Cancel the common factor of $d^{6}$ and $d^{20}$ .

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Step 4.1

Multiply by $1$ .

$\frac{d^{6} \cdot 1}{d^{20} x^{2} \cdot 324}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $d^{6}$ out of $d^{20} x^{2} \cdot 324$ .

$\frac{d^{6} \cdot 1}{d^{6} (d^{14} x^{2} \cdot 324)}$

Step 4.2.2

Cancel the common factor.

$\frac{d^{6} \cdot 1}{d^{6} (d^{14} x^{2} \cdot 324)}$

Step 4.2.3

Rewrite the expression.

$\frac{1}{d^{14} x^{2} \cdot 324}$

Step 5

Move $324$ to the left of $d^{14} x^{2}$ .

$\frac{1}{324 d^{14} x^{2}}$