Algebra Examples

$\frac{{(4 v w^{- 3} x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $4 v w^{- 3} x^{2}$ .

$\frac{{(4 v w^{- 3})}^{2} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.2

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

$\frac{{(4 v \frac{1}{w^{3}})}^{2} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.3

Multiply $4 v \frac{1}{w^{3}}$ .

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Step 1.3.1

Combine $4$ and $\frac{1}{w^{3}}$ .

$\frac{{(v \frac{4}{w^{3}})}^{2} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.3.2

Combine $v$ and $\frac{4}{w^{3}}$ .

$\frac{{(\frac{v \cdot 4}{w^{3}})}^{2} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.4

Move $4$ to the left of $v$ .

$\frac{{(\frac{4 \cdot v}{w^{3}})}^{2} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.5

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

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Step 1.5.1

Apply the product rule to $\frac{4 v}{w^{3}}$ .

$\frac{\frac{{(4 v)}^{2}}{{(w^{3})}^{2}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.5.2

Apply the product rule to $4 v$ .

$\frac{\frac{4^{2} v^{2}}{{(w^{3})}^{2}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.6

Raise $4$ to the power of $2$ .

$\frac{\frac{16 v^{2}}{{(w^{3})}^{2}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.7

Multiply the exponents in ${(w^{3})}^{2}$ .

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Step 1.7.1

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

$\frac{\frac{16 v^{2}}{w^{3 \cdot 2}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.7.2

Multiply $3$ by $2$ .

$\frac{\frac{16 v^{2}}{w^{6}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.8

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

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Step 1.8.1

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

$\frac{\frac{16 v^{2}}{w^{6}} x^{2 \cdot 2}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 1.8.2

Multiply $2$ by $2$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{{(2 v^{3} w^{2} x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $2 v^{3} w^{2} x^{- 2}$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{{(2 v^{3} w^{2})}^{4} {(x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.2

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

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Step 2.2.1

Apply the product rule to $2 v^{3} w^{2}$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{{(2 v^{3})}^{4} {(w^{2})}^{4} {(x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.2.2

Apply the product rule to $2 v^{3}$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{2^{4} {(v^{3})}^{4} {(w^{2})}^{4} {(x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.3

Raise $2$ to the power of $4$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{16 {(v^{3})}^{4} {(w^{2})}^{4} {(x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.4

Multiply the exponents in ${(v^{3})}^{4}$ .

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Step 2.4.1

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

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{16 v^{3 \cdot 4} {(w^{2})}^{4} {(x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.4.2

Multiply $3$ by $4$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{16 v^{12} {(w^{2})}^{4} {(x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.5

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

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Step 2.5.1

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

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{16 v^{12} w^{2 \cdot 4} {(x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.5.2

Multiply $2$ by $4$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{16 v^{12} w^{8} {(x^{- 2})}^{4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.6

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

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Step 2.6.1

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

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{16 v^{12} w^{8} x^{- 2 \cdot 4}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.6.2

Multiply $- 2$ by $4$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{16 v^{12} w^{8} x^{- 8}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.7

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

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{16 v^{12} w^{8} \frac{1}{x^{8}}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.8

Combine exponents.

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Step 2.8.1

Combine $16$ and $\frac{1}{x^{8}}$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{v^{12} w^{8} \frac{16}{x^{8}}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.8.2

Combine $v^{12}$ and $\frac{16}{x^{8}}$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{w^{8} \frac{v^{12} \cdot 16}{x^{8}}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.8.3

Combine $w^{8}$ and $\frac{v^{12} \cdot 16}{x^{8}}$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{\frac{w^{8} (v^{12} \cdot 16)}{x^{8}}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.9

Remove unnecessary parentheses.

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{\frac{w^{8} v^{12} \cdot 16}{x^{8}}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 2.10

Move $16$ to the left of $w^{8} v^{12}$ .

$\frac{\frac{16 v^{2}}{w^{6}} x^{4}}{\frac{16 w^{8} v^{12}}{x^{8}}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 3

Combine $\frac{16 v^{2}}{w^{6}}$ and $x^{4}$ .

$\frac{\frac{16 v^{2} x^{4}}{w^{6}}}{\frac{16 w^{8} v^{12}}{x^{8}}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 v^{2} x^{4}}{w^{6}} \cdot \frac{x^{8}}{16 w^{8} v^{12}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 5

Combine.

$\frac{16 v^{2} x^{4} x^{8}}{w^{6} (16 w^{8} v^{12})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 6

Multiply $x^{4}$ by $x^{8}$ by adding the exponents.

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Step 6.1

Move $x^{8}$ .

$\frac{16 v^{2} (x^{8} x^{4})}{w^{6} (16 w^{8} v^{12})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 6.2

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

$\frac{16 v^{2} x^{8 + 4}}{w^{6} (16 w^{8} v^{12})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 6.3

Add $8$ and $4$ .

$\frac{16 v^{2} x^{12}}{w^{6} (16 w^{8} v^{12})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 7

Multiply $w^{6}$ by $w^{8}$ by adding the exponents.

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Step 7.1

Move $w^{8}$ .

$\frac{16 v^{2} x^{12}}{w^{8} w^{6} (16 v^{12})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 7.2

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

$\frac{16 v^{2} x^{12}}{w^{8 + 6} (16 v^{12})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 7.3

Add $8$ and $6$ .

$\frac{16 v^{2} x^{12}}{w^{14} (16 v^{12})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 8

Cancel the common factor of $16$ .

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Step 8.1

Cancel the common factor.

$\frac{16 v^{2} x^{12}}{w^{14} \cdot 16 v^{12}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 8.2

Rewrite the expression.

$\frac{v^{2} x^{12}}{w^{14} v^{12}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 9

Cancel the common factor of $v^{2}$ and $v^{12}$ .

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Step 9.1

Factor $v^{2}$ out of $v^{2} x^{12}$ .

$\frac{v^{2} (x^{12})}{w^{14} v^{12}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $v^{2}$ out of $w^{14} v^{12}$ .

$\frac{v^{2} (x^{12})}{v^{2} (w^{14} v^{10})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 9.2.2

Cancel the common factor.

$\frac{v^{2} x^{12}}{v^{2} (w^{14} v^{10})} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 9.2.3

Rewrite the expression.

$\frac{x^{12}}{w^{14} v^{10}} \cdot {(- v^{3} w^{2} x^{- 4})}^{- 3}$

Step 10

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

$\frac{x^{12}}{w^{14} v^{10}} \cdot {(- v^{3} w^{2} \frac{1}{x^{4}})}^{- 3}$

Step 11

Multiply $- v^{3} w^{2} \frac{1}{x^{4}}$ .

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Step 11.1

Combine $v^{3}$ and $\frac{1}{x^{4}}$ .

$\frac{x^{12}}{w^{14} v^{10}} \cdot {(- 1 w^{2} \frac{v^{3}}{x^{4}})}^{- 3}$

Step 11.2

Combine $\frac{v^{3}}{x^{4}}$ and $w^{2}$ .

$\frac{x^{12}}{w^{14} v^{10}} \cdot {(- 1 \frac{v^{3} w^{2}}{x^{4}})}^{- 3}$

Step 12

Rewrite $- 1 \frac{v^{3} w^{2}}{x^{4}}$ as $- \frac{v^{3} w^{2}}{x^{4}}$ .

$\frac{x^{12}}{w^{14} v^{10}} \cdot {(- \frac{v^{3} w^{2}}{x^{4}})}^{- 3}$

Step 13

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

$\frac{x^{12}}{w^{14} v^{10}} \cdot {(- \frac{x^{4}}{v^{3} w^{2}})}^{3}$

Step 14

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

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Step 14.1

Apply the product rule to $- \frac{x^{4}}{v^{3} w^{2}}$ .

$\frac{x^{12}}{w^{14} v^{10}} \cdot ({(- 1)}^{3} {(\frac{x^{4}}{v^{3} w^{2}})}^{3})$

Step 14.2

Apply the product rule to $\frac{x^{4}}{v^{3} w^{2}}$ .

$\frac{x^{12}}{w^{14} v^{10}} \cdot ({(- 1)}^{3} \frac{{(x^{4})}^{3}}{{(v^{3} w^{2})}^{3}})$

Step 14.3

Apply the product rule to $v^{3} w^{2}$ .

$\frac{x^{12}}{w^{14} v^{10}} \cdot ({(- 1)}^{3} \frac{{(x^{4})}^{3}}{{(v^{3})}^{3} {(w^{2})}^{3}})$

Step 15

Simplify the expression.

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Step 15.1

Rewrite using the commutative property of multiplication.

${(- 1)}^{3} \frac{x^{12}}{w^{14} v^{10}} \frac{{(x^{4})}^{3}}{{(v^{3})}^{3} {(w^{2})}^{3}}$

Step 15.2

Raise $- 1$ to the power of $3$ .

$- \frac{x^{12}}{w^{14} v^{10}} \cdot \frac{{(x^{4})}^{3}}{{(v^{3})}^{3} {(w^{2})}^{3}}$

Step 15.3

Multiply the exponents in ${(x^{4})}^{3}$ .

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Step 15.3.1

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

$- \frac{x^{12}}{w^{14} v^{10}} \cdot \frac{x^{4 \cdot 3}}{{(v^{3})}^{3} {(w^{2})}^{3}}$

Step 15.3.2

Multiply $4$ by $3$ .

$- \frac{x^{12}}{w^{14} v^{10}} \cdot \frac{x^{12}}{{(v^{3})}^{3} {(w^{2})}^{3}}$

Step 16

Simplify the denominator.

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Step 16.1

Multiply the exponents in ${(v^{3})}^{3}$ .

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Step 16.1.1

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

$- \frac{x^{12}}{w^{14} v^{10}} \cdot \frac{x^{12}}{v^{3 \cdot 3} {(w^{2})}^{3}}$

Step 16.1.2

Multiply $3$ by $3$ .

$- \frac{x^{12}}{w^{14} v^{10}} \cdot \frac{x^{12}}{v^{9} {(w^{2})}^{3}}$

Step 16.2

Multiply the exponents in ${(w^{2})}^{3}$ .

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Step 16.2.1

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

$- \frac{x^{12}}{w^{14} v^{10}} \cdot \frac{x^{12}}{v^{9} w^{2 \cdot 3}}$

Step 16.2.2

Multiply $2$ by $3$ .

$- \frac{x^{12}}{w^{14} v^{10}} \cdot \frac{x^{12}}{v^{9} w^{6}}$

Step 17

Multiply $- \frac{x^{12}}{w^{14} v^{10}} \cdot \frac{x^{12}}{v^{9} w^{6}}$ .

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Step 17.1

Multiply $\frac{x^{12}}{v^{9} w^{6}}$ by $\frac{x^{12}}{w^{14} v^{10}}$ .

$- \frac{x^{12} x^{12}}{v^{9} w^{6} (w^{14} v^{10})}$

Step 17.2

Multiply $x^{12}$ by $x^{12}$ by adding the exponents.

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Step 17.2.1

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

$- \frac{x^{12 + 12}}{v^{9} w^{6} (w^{14} v^{10})}$

Step 17.2.2

Add $12$ and $12$ .

$- \frac{x^{24}}{v^{9} w^{6} (w^{14} v^{10})}$

Step 17.3

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

$- \frac{x^{24}}{v^{10 + 9} w^{6} w^{14}}$

Step 17.4

Add $10$ and $9$ .

$- \frac{x^{24}}{v^{19} w^{6} w^{14}}$

Step 17.5

Multiply $w^{6}$ by $w^{14}$ by adding the exponents.

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Step 17.5.1

Move $w^{14}$ .

$- \frac{x^{24}}{v^{19} (w^{14} w^{6})}$

Step 17.5.2

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

$- \frac{x^{24}}{v^{19} w^{14 + 6}}$

Step 17.5.3

Add $14$ and $6$ .

$- \frac{x^{24}}{v^{19} w^{20}}$