Algebra Examples

${(\frac{2 m^{3} n^{- 5}}{m^{9} n^{- 3}})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 1

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

${(\frac{2 m^{3}}{m^{9} n^{- 3} n^{5}})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 2

Multiply $n^{- 3}$ by $n^{5}$ by adding the exponents.

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Step 2.1

Move $n^{5}$ .

${(\frac{2 m^{3}}{m^{9} (n^{5} n^{- 3})})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 2.2

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

${(\frac{2 m^{3}}{m^{9} n^{5 - 3}})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 2.3

Subtract $3$ from $5$ .

${(\frac{2 m^{3}}{m^{9} n^{2}})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 3

Cancel the common factor of $m^{3}$ and $m^{9}$ .

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Step 3.1

Factor $m^{3}$ out of $2 m^{3}$ .

${(\frac{m^{3} \cdot 2}{m^{9} n^{2}})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $m^{3}$ out of $m^{9} n^{2}$ .

${(\frac{m^{3} \cdot 2}{m^{3} (m^{6} n^{2})})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 3.2.2

Cancel the common factor.

${(\frac{m^{3} \cdot 2}{m^{3} (m^{6} n^{2})})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 3.2.3

Rewrite the expression.

${(\frac{2}{m^{6} n^{2}})}^{3} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{2}{m^{6} n^{2}}$ .

$\frac{2^{3}}{{(m^{6} n^{2})}^{3}} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 4.2

Apply the product rule to $m^{6} n^{2}$ .

$\frac{2^{3}}{{(m^{6})}^{3} {(n^{2})}^{3}} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 5

Raise $2$ to the power of $3$ .

$\frac{8}{{(m^{6})}^{3} {(n^{2})}^{3}} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 6

Simplify the denominator.

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Step 6.1

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

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Step 6.1.1

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

$\frac{8}{m^{6 \cdot 3} {(n^{2})}^{3}} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 6.1.2

Multiply $6$ by $3$ .

$\frac{8}{m^{18} {(n^{2})}^{3}} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 6.2

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

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Step 6.2.1

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

$\frac{8}{m^{18} n^{2 \cdot 3}} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 6.2.2

Multiply $2$ by $3$ .

$\frac{8}{m^{18} n^{6}} {(\frac{m^{5} n^{3}}{m^{- 1} n^{- 3}})}^{6}$

Step 7

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

$\frac{8}{m^{18} n^{6}} {(\frac{m^{5} n^{3} m}{n^{- 3}})}^{6}$

Step 8

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

$\frac{8}{m^{18} n^{6}} {(m^{5} n^{3} m n^{3})}^{6}$

Step 9

Multiply $m^{5}$ by $m$ by adding the exponents.

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Step 9.1

Move $m$ .

$\frac{8}{m^{18} n^{6}} {(m \cdot m^{5} n^{3} n^{3})}^{6}$

Step 9.2

Multiply $m$ by $m^{5}$ .

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Step 9.2.1

Raise $m$ to the power of $1$ .

$\frac{8}{m^{18} n^{6}} {(m^{1} m^{5} n^{3} n^{3})}^{6}$

Step 9.2.2

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

$\frac{8}{m^{18} n^{6}} {(m^{1 + 5} n^{3} n^{3})}^{6}$

Step 9.3

Add $1$ and $5$ .

$\frac{8}{m^{18} n^{6}} {(m^{6} n^{3} n^{3})}^{6}$

Step 10

Multiply $n^{3}$ by $n^{3}$ by adding the exponents.

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Step 10.1

Move $n^{3}$ .

$\frac{8}{m^{18} n^{6}} {(m^{6} (n^{3} n^{3}))}^{6}$

Step 10.2

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

$\frac{8}{m^{18} n^{6}} {(m^{6} n^{3 + 3})}^{6}$

Step 10.3

Add $3$ and $3$ .

$\frac{8}{m^{18} n^{6}} {(m^{6} n^{6})}^{6}$

Step 11

Apply basic rules of exponents.

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Step 11.1

Apply the product rule to $m^{6} n^{6}$ .

$\frac{8}{m^{18} n^{6}} ({(m^{6})}^{6} {(n^{6})}^{6})$

Step 11.2

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

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Step 11.2.1

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

$\frac{8}{m^{18} n^{6}} (m^{6 \cdot 6} {(n^{6})}^{6})$

Step 11.2.2

Multiply $6$ by $6$ .

$\frac{8}{m^{18} n^{6}} (m^{36} {(n^{6})}^{6})$

Step 12

Cancel the common factor of $m^{18}$ .

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Step 12.1

Factor $m^{18}$ out of $m^{18} n^{6}$ .

$\frac{8}{m^{18} (n^{6})} (m^{36} {(n^{6})}^{6})$

Step 12.2

Factor $m^{18}$ out of $m^{36} {(n^{6})}^{6}$ .

$\frac{8}{m^{18} (n^{6})} (m^{18} (m^{18} {(n^{6})}^{6}))$

Step 12.3

Cancel the common factor.

$\frac{8}{m^{18} n^{6}} (m^{18} (m^{18} {(n^{6})}^{6}))$

Step 12.4

Rewrite the expression.

$\frac{8}{n^{6}} (m^{18} {(n^{6})}^{6})$

Step 13

Combine $m^{18}$ and $\frac{8}{n^{6}}$ .

$\frac{m^{18} \cdot 8}{n^{6}} {(n^{6})}^{6}$

Step 14

Combine $\frac{m^{18} \cdot 8}{n^{6}}$ and ${(n^{6})}^{6}$ .

$\frac{m^{18} \cdot 8 {(n^{6})}^{6}}{n^{6}}$

Step 15

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

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Step 15.1

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

$\frac{m^{18} \cdot 8 n^{6 \cdot 6}}{n^{6}}$

Step 15.2

Multiply $6$ by $6$ .

$\frac{m^{18} \cdot 8 n^{36}}{n^{6}}$

Step 16

Cancel the common factor of $n^{36}$ and $n^{6}$ .

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Step 16.1

Factor $n^{6}$ out of $m^{18} \cdot 8 n^{36}$ .

$\frac{n^{6} (m^{18} \cdot 8 n^{30})}{n^{6}}$

Step 16.2

Cancel the common factors.

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Step 16.2.1

Multiply by $1$ .

$\frac{n^{6} (m^{18} \cdot 8 n^{30})}{n^{6} \cdot 1}$

Step 16.2.2

Cancel the common factor.

$\frac{n^{6} (m^{18} \cdot 8 n^{30})}{n^{6} \cdot 1}$

Step 16.2.3

Rewrite the expression.

$\frac{m^{18} \cdot 8 n^{30}}{1}$

Step 16.2.4

Divide $m^{18} \cdot 8 n^{30}$ by $1$ .

$m^{18} \cdot 8 n^{30}$

Step 17

Move $8$ to the left of $m^{18}$ .

$8 m^{18} n^{30}$