Basic Math Examples

(v w^{5}) {(\frac{u^{- 1} v^{2}}{2 w^{2}})}^{- 3}

$(v w^{5}) {(\frac{u^{- 1} v^{2}}{2 w^{2}})}^{- 3}$

Step 1

Move

u^{- 1}

$u^{- 1}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

v w^{5} {(\frac{v^{2}}{2 w^{2} u})}^{- 3}

$v w^{5} {(\frac{v^{2}}{2 w^{2} u})}^{- 3}$

Step 2

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

v w^{5} {(\frac{2 w^{2} u}{v^{2}})}^{3}

$v w^{5} {(\frac{2 w^{2} u}{v^{2}})}^{3}$

Step 3

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

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

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Step 3.1

Apply the product rule to

\frac{2 w^{2} u}{v^{2}}

$\frac{2 w^{2} u}{v^{2}}$ .

v w^{5} \frac{{(2 w^{2} u)}^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{{(2 w^{2} u)}^{3}}{{(v^{2})}^{3}}$

Step 3.2

Apply the product rule to

2 w^{2} u

$2 w^{2} u$ .

v w^{5} \frac{{(2 w^{2})}^{3} u^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{{(2 w^{2})}^{3} u^{3}}{{(v^{2})}^{3}}$

Step 3.3

Apply the product rule to

2 w^{2}

$2 w^{2}$ .

v w^{5} \frac{2^{3} {(w^{2})}^{3} u^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{2^{3} {(w^{2})}^{3} u^{3}}{{(v^{2})}^{3}}$

v w^{5} \frac{2^{3} {(w^{2})}^{3} u^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{2^{3} {(w^{2})}^{3} u^{3}}{{(v^{2})}^{3}}$

Step 4

Simplify the numerator.

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Step 4.1

Raise

2

$2$ to the power of

3

$3$ .

v w^{5} \frac{8 {(w^{2})}^{3} u^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{8 {(w^{2})}^{3} u^{3}}{{(v^{2})}^{3}}$

Step 4.2

Multiply the exponents in

{(w^{2})}^{3}

${(w^{2})}^{3}$ .

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Step 4.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

v w^{5} \frac{8 w^{2 \cdot 3} u^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{8 w^{2 \cdot 3} u^{3}}{{(v^{2})}^{3}}$

Step 4.2.2

Multiply

2

$2$ by

3

$3$ .

v w^{5} \frac{8 w^{6} u^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{8 w^{6} u^{3}}{{(v^{2})}^{3}}$

v w^{5} \frac{8 w^{6} u^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{8 w^{6} u^{3}}{{(v^{2})}^{3}}$

v w^{5} \frac{8 w^{6} u^{3}}{{(v^{2})}^{3}}

$v w^{5} \frac{8 w^{6} u^{3}}{{(v^{2})}^{3}}$

Step 5

Simplify terms.

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Step 5.1

Multiply the exponents in

{(v^{2})}^{3}

${(v^{2})}^{3}$ .

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Step 5.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

v w^{5} \frac{8 w^{6} u^{3}}{v^{2 \cdot 3}}

$v w^{5} \frac{8 w^{6} u^{3}}{v^{2 \cdot 3}}$

Step 5.1.2

Multiply

2

$2$ by

3

$3$ .

v w^{5} \frac{8 w^{6} u^{3}}{v^{6}}

$v w^{5} \frac{8 w^{6} u^{3}}{v^{6}}$

v w^{5} \frac{8 w^{6} u^{3}}{v^{6}}

$v w^{5} \frac{8 w^{6} u^{3}}{v^{6}}$

Step 5.2

Cancel the common factor of

v

$v$ .

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Step 5.2.1

Factor

v

$v$ out of

v w^{5}

$v w^{5}$ .

v (w^{5}) \frac{8 w^{6} u^{3}}{v^{6}}

$v (w^{5}) \frac{8 w^{6} u^{3}}{v^{6}}$

Step 5.2.2

Factor

v

$v$ out of

v^{6}

$v^{6}$ .

v (w^{5}) \frac{8 w^{6} u^{3}}{v \cdot v^{5}}

$v (w^{5}) \frac{8 w^{6} u^{3}}{v \cdot v^{5}}$

Step 5.2.3

Cancel the common factor.

$v w^{5} \frac{8 w^{6} u^{3}}{v \cdot v^{5}}$

Step 5.2.4

Rewrite the expression.

$w^{5} \frac{8 w^{6} u^{3}}{v^{5}}$

Step 5.3

Combine

$w^{5}$ and

$\frac{8 w^{6} u^{3}}{v^{5}}$ .

$\frac{w^{5} (8 w^{6} u^{3})}{v^{5}}$

Step 6

Multiply

$w^{5}$ by

$w^{6}$ by adding the exponents.

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Step 6.1

Move

$w^{6}$ .

$\frac{w^{6} w^{5} (8 u^{3})}{v^{5}}$

Step 6.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{w^{6 + 5} (8 u^{3})}{v^{5}}$

Step 6.3

Add

$6$ and

$5$ .

$\frac{w^{11} (8 u^{3})}{v^{5}}$

Step 7

Move

$8$ to the left of

$w^{11}$ .

$\frac{8 w^{11} u^{3}}{v^{5}}$