Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $8 v^{2} w^{- 3} x^{2}$ .

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

Step 1.2

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

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

Step 1.3

Multiply $8 v^{2} \frac{1}{w^{3}}$ .

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Step 1.3.1

Combine $8$ and $\frac{1}{w^{3}}$ .

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

Step 1.3.2

Combine $v^{2}$ and $\frac{8}{w^{3}}$ .

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

Step 1.4

Move $8$ to the left of $v^{2}$ .

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

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 $\frac{8 v^{2}}{w^{3}}$ .

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

Step 1.5.2

Apply the product rule to $8 v^{2}$ .

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Raise $8$ to the power of $2$ .

$\frac{\frac{64 {(v^{2})}^{2}}{{(w^{3})}^{2}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 1.6.2

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

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Step 1.6.2.1

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

$\frac{\frac{64 v^{2 \cdot 2}}{{(w^{3})}^{2}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 1.6.2.2

Multiply $2$ by $2$ .

$\frac{\frac{64 v^{4}}{{(w^{3})}^{2}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

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

$\frac{\frac{64 v^{4}}{w^{3 \cdot 2}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 1.7.2

Multiply $3$ by $2$ .

$\frac{\frac{64 v^{4}}{w^{6}} {(x^{2})}^{2}}{{(2 v^{3} w^{2} x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

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

$\frac{\frac{64 v^{4}}{w^{6}} x^{2 \cdot 2}}{{(2 v^{3} w^{2} x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 1.8.2

Multiply $2$ by $2$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{{(2 v^{3} w^{2} x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{{(2 v^{3} w^{2})}^{6} {(x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

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 $2 v^{3} w^{2}$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{{(2 v^{3})}^{6} {(w^{2})}^{6} {(x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.2.2

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

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{2^{6} {(v^{3})}^{6} {(w^{2})}^{6} {(x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.3

Raise $2$ to the power of $6$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{64 {(v^{3})}^{6} {(w^{2})}^{6} {(x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.4

Multiply the exponents in ${(v^{3})}^{6}$ .

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Step 2.4.1

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

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{64 v^{3 \cdot 6} {(w^{2})}^{6} {(x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.4.2

Multiply $3$ by $6$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{64 v^{18} {(w^{2})}^{6} {(x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.5

Multiply the exponents in ${(w^{2})}^{6}$ .

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Step 2.5.1

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

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{64 v^{18} w^{2 \cdot 6} {(x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.5.2

Multiply $2$ by $6$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{64 v^{18} w^{12} {(x^{- 4})}^{6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.6

Multiply the exponents in ${(x^{- 4})}^{6}$ .

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Step 2.6.1

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

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{64 v^{18} w^{12} x^{- 4 \cdot 6}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.6.2

Multiply $- 4$ by $6$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{64 v^{18} w^{12} x^{- 24}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.7

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

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{64 v^{18} w^{12} \frac{1}{x^{24}}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.8

Combine exponents.

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Step 2.8.1

Combine $64$ and $\frac{1}{x^{24}}$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{v^{18} w^{12} \frac{64}{x^{24}}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.8.2

Combine $v^{18}$ and $\frac{64}{x^{24}}$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{w^{12} \frac{v^{18} \cdot 64}{x^{24}}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.8.3

Combine $w^{12}$ and $\frac{v^{18} \cdot 64}{x^{24}}$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{\frac{w^{12} (v^{18} \cdot 64)}{x^{24}}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.9

Remove unnecessary parentheses.

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{\frac{w^{12} v^{18} \cdot 64}{x^{24}}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 2.10

Move $64$ to the left of $w^{12} v^{18}$ .

$\frac{\frac{64 v^{4}}{w^{6}} x^{4}}{\frac{64 w^{12} v^{18}}{x^{24}}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 3

Combine $\frac{64 v^{4}}{w^{6}}$ and $x^{4}$ .

$\frac{\frac{64 v^{4} x^{4}}{w^{6}}}{\frac{64 w^{12} v^{18}}{x^{24}}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{64 v^{4} x^{4}}{w^{6}} \cdot \frac{x^{24}}{64 w^{12} v^{18}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 5

Combine.

$\frac{64 v^{4} x^{4} x^{24}}{w^{6} (64 w^{12} v^{18})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 6

Multiply $x^{4}$ by $x^{24}$ by adding the exponents.

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Step 6.1

Move $x^{24}$ .

$\frac{64 v^{4} (x^{24} x^{4})}{w^{6} (64 w^{12} v^{18})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 6.2

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

$\frac{64 v^{4} x^{24 + 4}}{w^{6} (64 w^{12} v^{18})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 6.3

Add $24$ and $4$ .

$\frac{64 v^{4} x^{28}}{w^{6} (64 w^{12} v^{18})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 7

Multiply $w^{6}$ by $w^{12}$ by adding the exponents.

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Step 7.1

Move $w^{12}$ .

$\frac{64 v^{4} x^{28}}{w^{12} w^{6} (64 v^{18})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 7.2

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

$\frac{64 v^{4} x^{28}}{w^{12 + 6} (64 v^{18})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 7.3

Add $12$ and $6$ .

$\frac{64 v^{4} x^{28}}{w^{18} (64 v^{18})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 8

Cancel the common factor of $64$ .

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Step 8.1

Cancel the common factor.

$\frac{64 v^{4} x^{28}}{w^{18} \cdot 64 v^{18}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 8.2

Rewrite the expression.

$\frac{v^{4} x^{28}}{w^{18} v^{18}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 9

Cancel the common factor of $v^{4}$ and $v^{18}$ .

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Step 9.1

Factor $v^{4}$ out of $v^{4} x^{28}$ .

$\frac{v^{4} (x^{28})}{w^{18} v^{18}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $v^{4}$ out of $w^{18} v^{18}$ .

$\frac{v^{4} (x^{28})}{v^{4} (w^{18} v^{14})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 9.2.2

Cancel the common factor.

$\frac{v^{4} x^{28}}{v^{4} (w^{18} v^{14})} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 9.2.3

Rewrite the expression.

$\frac{x^{28}}{w^{18} v^{14}} \cdot {(- v^{2} w^{3} x^{- 2})}^{- 3}$

Step 10

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

$\frac{x^{28}}{w^{18} v^{14}} \cdot {(- v^{2} w^{3} \frac{1}{x^{2}})}^{- 3}$

Step 11

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

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Step 11.1

Combine $v^{2}$ and $\frac{1}{x^{2}}$ .

$\frac{x^{28}}{w^{18} v^{14}} \cdot {(- 1 w^{3} \frac{v^{2}}{x^{2}})}^{- 3}$

Step 11.2

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

$\frac{x^{28}}{w^{18} v^{14}} \cdot {(- 1 \frac{v^{2} w^{3}}{x^{2}})}^{- 3}$

Step 12

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

$\frac{x^{28}}{w^{18} v^{14}} \cdot {(- \frac{v^{2} w^{3}}{x^{2}})}^{- 3}$

Step 13

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

$\frac{x^{28}}{w^{18} v^{14}} \cdot {(- \frac{x^{2}}{v^{2} w^{3}})}^{3}$

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

$\frac{x^{28}}{w^{18} v^{14}} \cdot ({(- 1)}^{3} {(\frac{x^{2}}{v^{2} w^{3}})}^{3})$

Step 14.2

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

$\frac{x^{28}}{w^{18} v^{14}} \cdot ({(- 1)}^{3} \frac{{(x^{2})}^{3}}{{(v^{2} w^{3})}^{3}})$

Step 14.3

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

$\frac{x^{28}}{w^{18} v^{14}} \cdot ({(- 1)}^{3} \frac{{(x^{2})}^{3}}{{(v^{2})}^{3} {(w^{3})}^{3}})$

Step 15

Simplify the expression.

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Step 15.1

Rewrite using the commutative property of multiplication.

${(- 1)}^{3} \frac{x^{28}}{w^{18} v^{14}} \frac{{(x^{2})}^{3}}{{(v^{2})}^{3} {(w^{3})}^{3}}$

Step 15.2

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

$- \frac{x^{28}}{w^{18} v^{14}} \cdot \frac{{(x^{2})}^{3}}{{(v^{2})}^{3} {(w^{3})}^{3}}$

Step 15.3

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

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Step 15.3.1

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

$- \frac{x^{28}}{w^{18} v^{14}} \cdot \frac{x^{2 \cdot 3}}{{(v^{2})}^{3} {(w^{3})}^{3}}$

Step 15.3.2

Multiply $2$ by $3$ .

$- \frac{x^{28}}{w^{18} v^{14}} \cdot \frac{x^{6}}{{(v^{2})}^{3} {(w^{3})}^{3}}$

Step 16

Simplify the denominator.

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Step 16.1

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

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Step 16.1.1

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

$- \frac{x^{28}}{w^{18} v^{14}} \cdot \frac{x^{6}}{v^{2 \cdot 3} {(w^{3})}^{3}}$

Step 16.1.2

Multiply $2$ by $3$ .

$- \frac{x^{28}}{w^{18} v^{14}} \cdot \frac{x^{6}}{v^{6} {(w^{3})}^{3}}$

Step 16.2

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

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Step 16.2.1

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

$- \frac{x^{28}}{w^{18} v^{14}} \cdot \frac{x^{6}}{v^{6} w^{3 \cdot 3}}$

Step 16.2.2

Multiply $3$ by $3$ .

$- \frac{x^{28}}{w^{18} v^{14}} \cdot \frac{x^{6}}{v^{6} w^{9}}$

Step 17

Multiply $- \frac{x^{28}}{w^{18} v^{14}} \cdot \frac{x^{6}}{v^{6} w^{9}}$ .

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Step 17.1

Multiply $\frac{x^{6}}{v^{6} w^{9}}$ by $\frac{x^{28}}{w^{18} v^{14}}$ .

$- \frac{x^{6} x^{28}}{v^{6} w^{9} (w^{18} v^{14})}$

Step 17.2

Multiply $x^{6}$ by $x^{28}$ 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.

$- \frac{x^{6 + 28}}{v^{6} w^{9} (w^{18} v^{14})}$

Step 17.2.2

Add $6$ and $28$ .

$- \frac{x^{34}}{v^{6} w^{9} (w^{18} v^{14})}$

Step 17.3

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

$- \frac{x^{34}}{v^{14 + 6} w^{9} w^{18}}$

Step 17.4

Add $14$ and $6$ .

$- \frac{x^{34}}{v^{20} w^{9} w^{18}}$

Step 17.5

Multiply $w^{9}$ by $w^{18}$ by adding the exponents.

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Step 17.5.1

Move $w^{18}$ .

$- \frac{x^{34}}{v^{20} (w^{18} w^{9})}$

Step 17.5.2

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

$- \frac{x^{34}}{v^{20} w^{18 + 9}}$

Step 17.5.3

Add $18$ and $9$ .

$- \frac{x^{34}}{v^{20} w^{27}}$