Basic Math Examples

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

Step 1

Simplify.

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Step 1.1

Multiply $n$ by $1$ .

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

Step 1.2

Multiply $m$ by $1$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

Move $3$ to the left of $m$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Apply the product rule to $3 m$ .

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

Step 2.4

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

$\frac{\frac{9 m^{2}}{4^{2}} \cdot {(\frac{n}{4})}^{2}}{{(\frac{m}{4} \cdot n^{- \frac{1}{2}})}^{3}}$

Step 2.5

Raise $4$ to the power of $2$ .

$\frac{\frac{9 m^{2}}{16} \cdot {(\frac{n}{4})}^{2}}{{(\frac{m}{4} \cdot n^{- \frac{1}{2}})}^{3}}$

Step 2.6

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

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{4^{2}}}{{(\frac{m}{4} \cdot n^{- \frac{1}{2}})}^{3}}$

Step 2.7

Raise $4$ to the power of $2$ .

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{{(\frac{m}{4} \cdot n^{- \frac{1}{2}})}^{3}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $\frac{m}{4} \cdot n^{- \frac{1}{2}}$ .

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{{(\frac{m}{4})}^{3} \cdot {(n^{- \frac{1}{2}})}^{3}}$

Step 3.2

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

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{\frac{m^{3}}{4^{3}} \cdot {(n^{- \frac{1}{2}})}^{3}}$

Step 3.3

Raise $4$ to the power of $3$ .

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{\frac{m^{3}}{64} \cdot {(n^{- \frac{1}{2}})}^{3}}$

Step 3.4

Multiply the exponents in ${(n^{- \frac{1}{2}})}^{3}$ .

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Step 3.4.1

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

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{\frac{m^{3}}{64} \cdot n^{- \frac{1}{2} \cdot 3}}$

Step 3.4.2

Multiply $- \frac{1}{2} \cdot 3$ .

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Step 3.4.2.1

Multiply $3$ by $- 1$ .

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{\frac{m^{3}}{64} \cdot n^{- 3 (\frac{1}{2})}}$

Step 3.4.2.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{\frac{m^{3}}{64} \cdot n^{\frac{- 3}{2}}}$

Step 3.4.3

Move the negative in front of the fraction.

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{\frac{m^{3}}{64} \cdot n^{- \frac{3}{2}}}$

Step 3.5

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

$\frac{\frac{9 m^{2}}{16} \cdot \frac{n^{2}}{16}}{\frac{m^{3}}{64} \cdot \frac{1}{n^{\frac{3}{2}}}}$

Step 4

Multiply $\frac{9 m^{2}}{16}$ by $\frac{n^{2}}{16}$ .

$\frac{\frac{9 m^{2} n^{2}}{16 \cdot 16}}{\frac{m^{3}}{64} \cdot \frac{1}{n^{\frac{3}{2}}}}$

Step 5

Multiply $\frac{m^{3}}{64}$ by $\frac{1}{n^{\frac{3}{2}}}$ .

$\frac{\frac{9 m^{2} n^{2}}{16 \cdot 16}}{\frac{m^{3}}{64 n^{\frac{3}{2}}}}$

Step 6

Multiply $16$ by $16$ .

$\frac{\frac{9 m^{2} n^{2}}{256}}{\frac{m^{3}}{64 n^{\frac{3}{2}}}}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 m^{2} n^{2}}{256} \cdot \frac{64 n^{\frac{3}{2}}}{m^{3}}$

Step 8

Combine.

$\frac{9 m^{2} n^{2} (64 n^{\frac{3}{2}})}{256 m^{3}}$

Step 9

Multiply $n^{2}$ by $n^{\frac{3}{2}}$ by adding the exponents.

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Step 9.1

Move $n^{\frac{3}{2}}$ .

$\frac{9 m^{2} (n^{\frac{3}{2}} n^{2}) \cdot 64}{256 m^{3}}$

Step 9.2

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

$\frac{9 m^{2} n^{\frac{3}{2} + 2} \cdot 64}{256 m^{3}}$

Step 9.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9 m^{2} n^{\frac{3}{2} + 2 \cdot \frac{2}{2}} \cdot 64}{256 m^{3}}$

Step 9.4

Combine $2$ and $\frac{2}{2}$ .

$\frac{9 m^{2} n^{\frac{3}{2} + \frac{2 \cdot 2}{2}} \cdot 64}{256 m^{3}}$

Step 9.5

Combine the numerators over the common denominator.

$\frac{9 m^{2} n^{\frac{3 + 2 \cdot 2}{2}} \cdot 64}{256 m^{3}}$

Step 9.6

Simplify the numerator.

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Step 9.6.1

Multiply $2$ by $2$ .

$\frac{9 m^{2} n^{\frac{3 + 4}{2}} \cdot 64}{256 m^{3}}$

Step 9.6.2

Add $3$ and $4$ .

$\frac{9 m^{2} n^{\frac{7}{2}} \cdot 64}{256 m^{3}}$

Step 10

Factor $m^{2}$ out of $9 m^{2} n^{\frac{7}{2}} \cdot 64$ .

$\frac{m^{2} (9 n^{\frac{7}{2}} \cdot 64)}{256 m^{3}}$

Step 11

Cancel the common factors.

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Step 11.1

Factor $m^{2}$ out of $256 m^{3}$ .

$\frac{m^{2} (9 n^{\frac{7}{2}} \cdot 64)}{m^{2} (256 m)}$

Step 11.2

Cancel the common factor.

$\frac{m^{2} (9 n^{\frac{7}{2}} \cdot 64)}{m^{2} (256 m)}$

Step 11.3

Rewrite the expression.

$\frac{9 n^{\frac{7}{2}} \cdot 64}{256 m}$

Step 12

Factor $64$ out of $9 n^{\frac{7}{2}} \cdot 64$ .

$\frac{64 \cdot (9 n^{\frac{7}{2}})}{256 m}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $64$ out of $256 m$ .

$\frac{64 \cdot (9 n^{\frac{7}{2}})}{64 (4 m)}$

Step 13.2

Cancel the common factor.

$\frac{64 \cdot (9 n^{\frac{7}{2}})}{64 (4 m)}$

Step 13.3

Rewrite the expression.

$\frac{9 n^{\frac{7}{2}}}{4 m}$