Algebra Examples

$\frac{5 b^{- 10} v^{- 10}}{{(3 b v^{- 1})}^{- 4}}$

Step 1

Move $b^{- 10}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{5 v^{- 10}}{{(3 b v^{- 1})}^{- 4} b^{10}}$

Step 2

Move $v^{- 10}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{5}{{(3 b v^{- 1})}^{- 4} b^{10} v^{10}}$

Step 3

Move ${(3 b v^{- 1})}^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{5 {(3 b v^{- 1})}^{4}}{b^{10} v^{10}}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the product rule to $3 b v^{- 1}$ .

$\frac{5 {(3 b)}^{4} {(v^{- 1})}^{4}}{b^{10} v^{10}}$

Step 4.2

Apply the product rule to $3 b$ .

$\frac{5 (3^{4} b^{4}) {(v^{- 1})}^{4}}{b^{10} v^{10}}$

Step 4.3

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

$\frac{5 (81 b^{4}) {(v^{- 1})}^{4}}{b^{10} v^{10}}$

Step 4.4

Multiply the exponents in ${(v^{- 1})}^{4}$ .

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Step 4.4.1

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

$\frac{5 (81 b^{4}) v^{- 1 \cdot 4}}{b^{10} v^{10}}$

Step 4.4.2

Multiply $- 1$ by $4$ .

$\frac{5 (81 b^{4}) v^{- 4}}{b^{10} v^{10}}$

Step 4.5

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

$\frac{5 \cdot 81 b^{4} \frac{1}{v^{4}}}{b^{10} v^{10}}$

Step 4.6

Combine exponents.

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Step 4.6.1

Multiply $5$ by $81$ .

$\frac{405 b^{4} \frac{1}{v^{4}}}{b^{10} v^{10}}$

Step 4.6.2

Combine $405$ and $\frac{1}{v^{4}}$ .

$\frac{b^{4} \frac{405}{v^{4}}}{b^{10} v^{10}}$

Step 4.6.3

Combine $b^{4}$ and $\frac{405}{v^{4}}$ .

$\frac{\frac{b^{4} \cdot 405}{v^{4}}}{b^{10} v^{10}}$

Step 4.7

Move $405$ to the left of $b^{4}$ .

$\frac{\frac{405 b^{4}}{v^{4}}}{b^{10} v^{10}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{405 b^{4}}{v^{4}} \cdot \frac{1}{b^{10} v^{10}}$

Step 6

Combine.

$\frac{405 b^{4} \cdot 1}{v^{4} (b^{10} v^{10})}$

Step 7

Multiply $v^{4}$ by $v^{10}$ by adding the exponents.

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Step 7.1

Move $v^{10}$ .

$\frac{405 b^{4} \cdot 1}{v^{10} v^{4} b^{10}}$

Step 7.2

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

$\frac{405 b^{4} \cdot 1}{v^{10 + 4} b^{10}}$

Step 7.3

Add $10$ and $4$ .

$\frac{405 b^{4} \cdot 1}{v^{14} b^{10}}$

Step 8

Multiply $405 b^{4}$ by $1$ .

$\frac{405 b^{4}}{v^{14} b^{10}}$

Step 9

Cancel the common factor of $b^{4}$ and $b^{10}$ .

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Step 9.1

Factor $b^{4}$ out of $405 b^{4}$ .

$\frac{b^{4} \cdot 405}{v^{14} b^{10}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $b^{4}$ out of $v^{14} b^{10}$ .

$\frac{b^{4} \cdot 405}{b^{4} (v^{14} b^{6})}$

Step 9.2.2

Cancel the common factor.

$\frac{b^{4} \cdot 405}{b^{4} (v^{14} b^{6})}$

Step 9.2.3

Rewrite the expression.

$\frac{405}{v^{14} b^{6}}$