Basic Math Examples

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

Step 1

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

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

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $m^{- 3} n^{5}$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $- 3$ by $2$ .

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

Step 2.4

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

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Multiply $5$ by $2$ .

$\frac{\frac{1}{m^{6}} n^{10} m^{3} {(n^{3})}^{3}}{3}$

Step 2.6

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

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Step 2.6.1

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

$\frac{\frac{1}{m^{6}} n^{10} m^{3} n^{3 \cdot 3}}{3}$

Step 2.6.2

Multiply $3$ by $3$ .

$\frac{\frac{1}{m^{6}} n^{10} m^{3} n^{9}}{3}$

Step 2.7

Combine exponents.

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Step 2.7.1

Combine $\frac{1}{m^{6}}$ and $n^{10}$ .

$\frac{\frac{n^{10}}{m^{6}} m^{3} n^{9}}{3}$

Step 2.7.2

Combine $\frac{n^{10}}{m^{6}}$ and $m^{3}$ .

$\frac{\frac{n^{10} m^{3}}{m^{6}} n^{9}}{3}$

Step 2.7.3

Combine $\frac{n^{10} m^{3}}{m^{6}}$ and $n^{9}$ .

$\frac{\frac{n^{10} m^{3} n^{9}}{m^{6}}}{3}$

Step 2.7.4

Multiply $n^{10}$ by $n^{9}$ by adding the exponents.

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Step 2.7.4.1

Move $n^{9}$ .

$\frac{\frac{n^{9} n^{10} m^{3}}{m^{6}}}{3}$

Step 2.7.4.2

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

$\frac{\frac{n^{9 + 10} m^{3}}{m^{6}}}{3}$

Step 2.7.4.3

Add $9$ and $10$ .

$\frac{\frac{n^{19} m^{3}}{m^{6}}}{3}$

Step 2.8

Reduce the expression $\frac{n^{19} m^{3}}{m^{6}}$ by cancelling the common factors.

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Step 2.8.1

Factor $m^{3}$ out of $n^{19} m^{3}$ .

$\frac{\frac{m^{3} n^{19}}{m^{6}}}{3}$

Step 2.8.2

Factor $m^{3}$ out of $m^{6}$ .

$\frac{\frac{m^{3} n^{19}}{m^{3} m^{3}}}{3}$

Step 2.8.3

Cancel the common factor.

$\frac{\frac{m^{3} n^{19}}{m^{3} m^{3}}}{3}$

Step 2.8.4

Rewrite the expression.

$\frac{\frac{n^{19}}{m^{3}}}{3}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{n^{19}}{m^{3}} \cdot \frac{1}{3}$

Step 4

Combine.

$\frac{n^{19} \cdot 1}{m^{3} \cdot 3}$

Step 5

Simplify the expression.

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Step 5.1

Multiply $n^{19}$ by $1$ .

$\frac{n^{19}}{m^{3} \cdot 3}$

Step 5.2

Move $3$ to the left of $m^{3}$ .

$\frac{n^{19}}{3 m^{3}}$