Basic Math Examples

${({(m^{7} n^{- 5})}^{3} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 1

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

${({(m^{7} \frac{1}{n^{5}})}^{3} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 2

Combine $m^{7}$ and $\frac{1}{n^{5}}$ .

${({(\frac{m^{7}}{n^{5}})}^{3} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 3

Apply basic rules of exponents.

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Step 3.1

Apply the product rule to $\frac{m^{7}}{n^{5}}$ .

${(\frac{{(m^{7})}^{3}}{{(n^{5})}^{3}} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 3.2

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

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Step 3.2.1

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

${(\frac{m^{7 \cdot 3}}{{(n^{5})}^{3}} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 3.2.2

Multiply $7$ by $3$ .

${(\frac{m^{21}}{{(n^{5})}^{3}} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 3.3

Multiply the exponents in ${(n^{5})}^{3}$ .

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Step 3.3.1

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

${(\frac{m^{21}}{n^{5 \cdot 3}} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 3.3.2

Multiply $5$ by $3$ .

${(\frac{m^{21}}{n^{15}} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 4

Cancel the common factor of $n^{- 7}$ .

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Step 4.1

Factor $n^{- 7}$ out of $n^{15}$ .

${(\frac{m^{21}}{n^{- 7} n^{22}} \cdot (m^{8} n^{- 7}))}^{- 1}$

Step 4.2

Factor $n^{- 7}$ out of $m^{8} n^{- 7}$ .

${(\frac{m^{21}}{n^{- 7} n^{22}} \cdot (n^{- 7} m^{8}))}^{- 1}$

Step 4.3

Cancel the common factor.

${(\frac{m^{21}}{n^{- 7} n^{22}} \cdot (n^{- 7} m^{8}))}^{- 1}$

Step 4.4

Rewrite the expression.

${(\frac{m^{21}}{n^{22}} \cdot m^{8})}^{- 1}$

Step 5

Combine $\frac{m^{21}}{n^{22}}$ and $m^{8}$ .

${(\frac{m^{21} m^{8}}{n^{22}})}^{- 1}$

Step 6

Multiply $m^{21}$ by $m^{8}$ by adding the exponents.

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Step 6.1

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

${(\frac{m^{21 + 8}}{n^{22}})}^{- 1}$

Step 6.2

Add $21$ and $8$ .

${(\frac{m^{29}}{n^{22}})}^{- 1}$

Step 7

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

$\frac{n^{22}}{m^{29}}$