Pre-Algebra Examples

${(\frac{m^{2} n^{- 3}}{3 n^{4} m^{- 2}})}^{- 3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 1

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

${(\frac{m^{2}}{3 n^{4} m^{- 2} n^{3}})}^{- 3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 2

Multiply $n^{4}$ by $n^{3}$ by adding the exponents.

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Step 2.1

Move $n^{3}$ .

${(\frac{m^{2}}{3 (n^{3} n^{4}) m^{- 2}})}^{- 3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 2.2

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

${(\frac{m^{2}}{3 n^{3 + 4} m^{- 2}})}^{- 3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 2.3

Add $3$ and $4$ .

${(\frac{m^{2}}{3 n^{7} m^{- 2}})}^{- 3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 3

Move $m^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{m^{2} m^{2}}{3 n^{7}})}^{- 3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 4

Multiply $m^{2}$ by $m^{2}$ by adding the exponents.

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Step 4.1

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

${(\frac{m^{2 + 2}}{3 n^{7}})}^{- 3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 4.2

Add $2$ and $2$ .

${(\frac{m^{4}}{3 n^{7}})}^{- 3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 5

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

${(\frac{3 n^{7}}{m^{4}})}^{3} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 6

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

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Step 6.1

Apply the product rule to $\frac{3 n^{7}}{m^{4}}$ .

$\frac{{(3 n^{7})}^{3}}{{(m^{4})}^{3}} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 6.2

Apply the product rule to $3 n^{7}$ .

$\frac{3^{3} {(n^{7})}^{3}}{{(m^{4})}^{3}} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 7

Simplify the numerator.

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Step 7.1

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

$\frac{27 {(n^{7})}^{3}}{{(m^{4})}^{3}} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 7.2

Multiply the exponents in ${(n^{7})}^{3}$ .

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Step 7.2.1

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

$\frac{27 n^{7 \cdot 3}}{{(m^{4})}^{3}} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 7.2.2

Multiply $7$ by $3$ .

$\frac{27 n^{21}}{{(m^{4})}^{3}} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

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

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Step 8.1.1

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

$\frac{27 n^{21}}{m^{4 \cdot 3}} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 8.1.2

Multiply $4$ by $3$ .

$\frac{27 n^{21}}{m^{12}} {(\frac{6 p^{3}}{3 p^{3} n^{- 5}})}^{- 2}$

Step 8.2

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

$\frac{27 n^{21}}{m^{12}} {(\frac{6 p^{3} n^{5}}{3 p^{3}})}^{- 2}$

Step 8.3

Cancel the common factor of $6$ and $3$ .

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Step 8.3.1

Factor $3$ out of $6 p^{3} n^{5}$ .

$\frac{27 n^{21}}{m^{12}} {(\frac{3 (2 p^{3} n^{5})}{3 p^{3}})}^{- 2}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $3$ out of $3 p^{3}$ .

$\frac{27 n^{21}}{m^{12}} {(\frac{3 (2 p^{3} n^{5})}{3 (p^{3})})}^{- 2}$

Step 8.3.2.2

Cancel the common factor.

$\frac{27 n^{21}}{m^{12}} {(\frac{3 (2 p^{3} n^{5})}{3 p^{3}})}^{- 2}$

Step 8.3.2.3

Rewrite the expression.

$\frac{27 n^{21}}{m^{12}} {(\frac{2 p^{3} n^{5}}{p^{3}})}^{- 2}$

Step 8.4

Cancel the common factor of $p^{3}$ .

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Step 8.4.1

Cancel the common factor.

$\frac{27 n^{21}}{m^{12}} {(\frac{2 p^{3} n^{5}}{p^{3}})}^{- 2}$

Step 8.4.2

Divide $2 n^{5}$ by $1$ .

$\frac{27 n^{21}}{m^{12}} {(2 n^{5})}^{- 2}$

Step 9

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

$\frac{27 n^{21}}{m^{12}} \cdot \frac{1}{{(2 n^{5})}^{2}}$

Step 10

Combine fractions.

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Step 10.1

Combine.

$\frac{27 n^{21} \cdot 1}{m^{12} {(2 n^{5})}^{2}}$

Step 10.2

Multiply $27$ by $1$ .

$\frac{27 n^{21}}{m^{12} {(2 n^{5})}^{2}}$

Step 11

Simplify the denominator.

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Step 11.1

Apply the product rule to $2 n^{5}$ .

$\frac{27 n^{21}}{m^{12} \cdot 2^{2} {(n^{5})}^{2}}$

Step 11.2

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

$\frac{27 n^{21}}{m^{12} \cdot 4 {(n^{5})}^{2}}$

Step 11.3

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

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Step 11.3.1

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

$\frac{27 n^{21}}{m^{12} \cdot 4 n^{5 \cdot 2}}$

Step 11.3.2

Multiply $5$ by $2$ .

$\frac{27 n^{21}}{m^{12} \cdot 4 n^{10}}$

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

Cancel the common factor of $n^{21}$ and $n^{10}$ .

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Step 12.1.1

Factor $n^{10}$ out of $27 n^{21}$ .

$\frac{n^{10} (27 n^{11})}{m^{12} \cdot 4 n^{10}}$

Step 12.1.2

Cancel the common factors.

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Step 12.1.2.1

Factor $n^{10}$ out of $m^{12} \cdot 4 n^{10}$ .

$\frac{n^{10} (27 n^{11})}{n^{10} (m^{12} \cdot 4)}$

Step 12.1.2.2

Cancel the common factor.

$\frac{n^{10} (27 n^{11})}{n^{10} (m^{12} \cdot 4)}$

Step 12.1.2.3

Rewrite the expression.

$\frac{27 n^{11}}{m^{12} \cdot 4}$

Step 12.2

Move $4$ to the left of $m^{12}$ .

$\frac{27 n^{11}}{4 m^{12}}$