Basic Math Examples

${(- 5 m^{- 1} n^{4})}^{3} {(n^{- 3} m^{- 2})}^{- 7}$

Step 1

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

${(- 5 \frac{1}{m} n^{4})}^{3} {(n^{- 3} m^{- 2})}^{- 7}$

Step 2

Combine $- 5$ and $\frac{1}{m}$ .

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

Step 3

Move the negative in front of the fraction.

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

Step 4

Combine $n^{4}$ and $\frac{5}{m}$ .

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

Step 5

Move $5$ to the left of $n^{4}$ .

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

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 $- \frac{5 n^{4}}{m}$ .

${(- 1)}^{3} {(\frac{5 n^{4}}{m})}^{3} {(n^{- 3} m^{- 2})}^{- 7}$

Step 6.2

Apply the product rule to $\frac{5 n^{4}}{m}$ .

${(- 1)}^{3} \frac{{(5 n^{4})}^{3}}{m^{3}} {(n^{- 3} m^{- 2})}^{- 7}$

Step 6.3

Apply the product rule to $5 n^{4}$ .

${(- 1)}^{3} \frac{5^{3} {(n^{4})}^{3}}{m^{3}} {(n^{- 3} m^{- 2})}^{- 7}$

Step 7

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

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

Step 8

Simplify the numerator.

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Step 8.1

Raise $5$ to the power of $3$ .

$- \frac{125 {(n^{4})}^{3}}{m^{3}} {(n^{- 3} m^{- 2})}^{- 7}$

Step 8.2

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

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Step 8.2.1

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

$- \frac{125 n^{4 \cdot 3}}{m^{3}} {(n^{- 3} m^{- 2})}^{- 7}$

Step 8.2.2

Multiply $4$ by $3$ .

$- \frac{125 n^{12}}{m^{3}} {(n^{- 3} m^{- 2})}^{- 7}$

Step 9

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

$- \frac{125 n^{12}}{m^{3}} {(\frac{1}{n^{3}} m^{- 2})}^{- 7}$

Step 10

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

$- \frac{125 n^{12}}{m^{3}} {(\frac{1}{n^{3}} \cdot \frac{1}{m^{2}})}^{- 7}$

Step 11

Combine.

$- \frac{125 n^{12}}{m^{3}} {(\frac{1 \cdot 1}{n^{3} m^{2}})}^{- 7}$

Step 12

Multiply $1$ by $1$ .

$- \frac{125 n^{12}}{m^{3}} {(\frac{1}{n^{3} m^{2}})}^{- 7}$

Step 13

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

$- \frac{125 n^{12}}{m^{3}} {(n^{3} m^{2})}^{7}$

Step 14

Apply basic rules of exponents.

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Step 14.1

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

$- \frac{125 n^{12}}{m^{3}} ({(n^{3})}^{7} {(m^{2})}^{7})$

Step 14.2

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

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Step 14.2.1

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

$- \frac{125 n^{12}}{m^{3}} (n^{3 \cdot 7} {(m^{2})}^{7})$

Step 14.2.2

Multiply $3$ by $7$ .

$- \frac{125 n^{12}}{m^{3}} (n^{21} {(m^{2})}^{7})$

Step 14.3

Multiply the exponents in ${(m^{2})}^{7}$ .

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Step 14.3.1

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

$- \frac{125 n^{12}}{m^{3}} (n^{21} m^{2 \cdot 7})$

Step 14.3.2

Multiply $2$ by $7$ .

$- \frac{125 n^{12}}{m^{3}} (n^{21} m^{14})$

Step 15

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

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Step 15.1

Move the leading negative in $- \frac{125 n^{12}}{m^{3}}$ into the numerator.

$\frac{- 125 n^{12}}{m^{3}} (n^{21} m^{14})$

Step 15.2

Factor $m^{3}$ out of $n^{21} m^{14}$ .

$\frac{- 125 n^{12}}{m^{3}} (m^{3} (n^{21} m^{11}))$

Step 15.3

Cancel the common factor.

$\frac{- 125 n^{12}}{m^{3}} (m^{3} (n^{21} m^{11}))$

Step 15.4

Rewrite the expression.

$- 125 n^{12} (n^{21} m^{11})$

Step 16

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

$- 125 n^{21 + 12} m^{11}$

Step 17

Add $21$ and $12$ .

$- 125 n^{33} m^{11}$