Algebra Examples

$\frac{10 v^{3} t^{- 5}}{{(2 v^{2} t^{- 2})}^{- 5}}$

Step 1

Simplify.

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Step 1.1

Move $t^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{10 v^{3}}{{(2 v^{2} t^{- 2})}^{- 5} t^{5}}$

Step 1.2

Move ${(2 v^{2} t^{- 2})}^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{10 v^{3} {(2 v^{2} t^{- 2})}^{5}}{t^{5}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $2 v^{2} t^{- 2}$ .

$\frac{10 v^{3} {(2 v^{2})}^{5} {(t^{- 2})}^{5}}{t^{5}}$

Step 2.2

Apply the product rule to $2 v^{2}$ .

$\frac{10 v^{3} (2^{5} {(v^{2})}^{5}) {(t^{- 2})}^{5}}{t^{5}}$

Step 2.3

Raise $2$ to the power of $5$ .

$\frac{10 v^{3} (32 {(v^{2})}^{5}) {(t^{- 2})}^{5}}{t^{5}}$

Step 2.4

Multiply the exponents in ${(v^{2})}^{5}$ .

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Step 2.4.1

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

$\frac{10 v^{3} (32 v^{2 \cdot 5}) {(t^{- 2})}^{5}}{t^{5}}$

Step 2.4.2

Multiply $2$ by $5$ .

$\frac{10 v^{3} (32 v^{10}) {(t^{- 2})}^{5}}{t^{5}}$

Step 2.5

Multiply the exponents in ${(t^{- 2})}^{5}$ .

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Step 2.5.1

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

$\frac{10 v^{3} (32 v^{10}) t^{- 2 \cdot 5}}{t^{5}}$

Step 2.5.2

Multiply $- 2$ by $5$ .

$\frac{10 v^{3} (32 v^{10}) t^{- 10}}{t^{5}}$

Step 2.6

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

$\frac{10 v^{3} \cdot 32 v^{10} \frac{1}{t^{10}}}{t^{5}}$

Step 2.7

Combine exponents.

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Step 2.7.1

Multiply $32$ by $10$ .

$\frac{320 v^{3} v^{10} \frac{1}{t^{10}}}{t^{5}}$

Step 2.7.2

Multiply $v^{3}$ by $v^{10}$ by adding the exponents.

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Step 2.7.2.1

Move $v^{10}$ .

$\frac{320 (v^{10} v^{3}) \frac{1}{t^{10}}}{t^{5}}$

Step 2.7.2.2

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

$\frac{320 v^{10 + 3} \frac{1}{t^{10}}}{t^{5}}$

Step 2.7.2.3

Add $10$ and $3$ .

$\frac{320 v^{13} \frac{1}{t^{10}}}{t^{5}}$

Step 2.7.3

Combine $320$ and $\frac{1}{t^{10}}$ .

$\frac{v^{13} \frac{320}{t^{10}}}{t^{5}}$

Step 2.7.4

Combine $v^{13}$ and $\frac{320}{t^{10}}$ .

$\frac{\frac{v^{13} \cdot 320}{t^{10}}}{t^{5}}$

Step 2.8

Move $320$ to the left of $v^{13}$ .

$\frac{\frac{320 v^{13}}{t^{10}}}{t^{5}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{320 v^{13}}{t^{10}} \cdot \frac{1}{t^{5}}$

Step 4

Combine.

$\frac{320 v^{13} \cdot 1}{t^{10} t^{5}}$

Step 5

Multiply $t^{10}$ by $t^{5}$ by adding the exponents.

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Step 5.1

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

$\frac{320 v^{13} \cdot 1}{t^{10 + 5}}$

Step 5.2

Add $10$ and $5$ .

$\frac{320 v^{13} \cdot 1}{t^{15}}$

Step 6

Multiply $320 v^{13}$ by $1$ .

$\frac{320 v^{13}}{t^{15}}$