Algebra Examples

$(\frac{3 m^{- 5} n^{2}}{4 m^{- 2} n^{0}}) \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{3 \frac{1}{m^{5}} n^{2}}{4 m^{- 2} n^{0}} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 1.2

Combine exponents.

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Step 1.2.1

Combine $3$ and $\frac{1}{m^{5}}$ .

$\frac{\frac{3}{m^{5}} n^{2}}{4 m^{- 2} n^{0}} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 1.2.2

Combine $\frac{3}{m^{5}}$ and $n^{2}$ .

$\frac{\frac{3 n^{2}}{m^{5}}}{4 m^{- 2} n^{0}} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{3 n^{2}}{m^{5}}}{4 \frac{1}{m^{2}} n^{0}} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 2.2

Anything raised to $0$ is $1$ .

$\frac{\frac{3 n^{2}}{m^{5}}}{4 \frac{1}{m^{2}} \cdot 1} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 2.3

Combine exponents.

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Step 2.3.1

Combine $4$ and $\frac{1}{m^{2}}$ .

$\frac{\frac{3 n^{2}}{m^{5}}}{\frac{4}{m^{2}} \cdot 1} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 2.3.2

Multiply $\frac{4}{m^{2}}$ by $1$ .

$\frac{\frac{3 n^{2}}{m^{5}}}{\frac{4}{m^{2}}} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 n^{2}}{m^{5}} \cdot \frac{m^{2}}{4} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 4

Combine.

$\frac{3 n^{2} m^{2}}{m^{5} \cdot 4} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 5

Cancel the common factor of $m^{2}$ and $m^{5}$ .

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Step 5.1

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

$\frac{m^{2} (3 n^{2})}{m^{5} \cdot 4} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $m^{2}$ out of $m^{5} \cdot 4$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

Move $4$ to the left of $m^{3}$ .

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

Step 7

Cancel the common factor of $n^{4}$ and $n$ .

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Step 7.1

Factor $n$ out of $m n^{4}$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $n$ out of $9 n$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

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

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Step 8.1

Apply the product rule to $\frac{m n^{3}}{9}$ .

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

Step 8.2

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

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

Step 9

Simplify terms.

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Step 9.1

Combine.

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

Step 9.2

Cancel the common factor of $m^{2}$ and $m^{3}$ .

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Step 9.2.1

Factor $m^{2}$ out of $3 n^{2} (m^{2} {(n^{3})}^{2})$ .

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

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $m^{2}$ out of $4 m^{3} \cdot 9^{2}$ .

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

Step 9.2.2.2

Cancel the common factor.

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

Step 9.2.2.3

Rewrite the expression.

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

Step 10

Simplify the numerator.

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

Multiply $3$ by $2$ .

$\frac{3 n^{2} n^{6}}{4 m \cdot 9^{2}}$

Step 10.2

Multiply $n^{2}$ by $n^{6}$ by adding the exponents.

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Step 10.2.1

Move $n^{6}$ .

$\frac{3 (n^{6} n^{2})}{4 m \cdot 9^{2}}$

Step 10.2.2

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

$\frac{3 n^{6 + 2}}{4 m \cdot 9^{2}}$

Step 10.2.3

Add $6$ and $2$ .

$\frac{3 n^{8}}{4 m \cdot 9^{2}}$

Step 11

Simplify the denominator.

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Step 11.1

Raise $9$ to the power of $2$ .

$\frac{3 n^{8}}{4 m \cdot 81}$

Step 11.2

Multiply $81$ by $4$ .

$\frac{3 n^{8}}{324 m}$

Step 12

Cancel the common factor of $3$ and $324$ .

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Step 12.1

Factor $3$ out of $3 n^{8}$ .

$\frac{3 (n^{8})}{324 m}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $3$ out of $324 m$ .

$\frac{3 (n^{8})}{3 (108 m)}$

Step 12.2.2

Cancel the common factor.

$\frac{3 n^{8}}{3 (108 m)}$

Step 12.2.3

Rewrite the expression.

$\frac{n^{8}}{108 m}$