Algebra Examples

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

Step 1

Move $m^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Multiply $m^{- 2}$ by $m^{5}$ by adding the exponents.

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Step 2.1.1

Move $m^{5}$ .

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

Step 2.1.2

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

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

Step 2.1.3

Subtract $2$ from $5$ .

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

Step 2.2

Simplify $4 m^{3} n^{0}$ .

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

Step 3

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

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Step 3.1

Apply the product rule to $\frac{3 n^{2}}{4 m^{3}}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $2$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $3$ by $2$ .

$\frac{9 n^{4}}{16 m^{6}} \cdot {(\frac{m n^{4}}{9 n})}^{2}$

Step 6

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

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Step 6.1

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

$\frac{9 n^{4}}{16 m^{6}} \cdot {(\frac{n (m n^{3})}{9 n})}^{2}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $n$ out of $9 n$ .

$\frac{9 n^{4}}{16 m^{6}} \cdot {(\frac{n (m n^{3})}{n \cdot 9})}^{2}$

Step 6.2.2

Cancel the common factor.

$\frac{9 n^{4}}{16 m^{6}} \cdot {(\frac{n (m n^{3})}{n \cdot 9})}^{2}$

Step 6.2.3

Rewrite the expression.

$\frac{9 n^{4}}{16 m^{6}} \cdot {(\frac{m n^{3}}{9})}^{2}$

Step 7

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

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Step 7.1

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

$\frac{9 n^{4}}{16 m^{6}} \cdot \frac{{(m n^{3})}^{2}}{9^{2}}$

Step 7.2

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

$\frac{9 n^{4}}{16 m^{6}} \cdot \frac{m^{2} {(n^{3})}^{2}}{9^{2}}$

Step 8

Combine.

$\frac{9 n^{4} (m^{2} {(n^{3})}^{2})}{16 m^{6} \cdot 9^{2}}$

Step 9

Cancel the common factor of $9$ and $9^{2}$ .

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Step 9.1

Factor $9$ out of $9 n^{4} (m^{2} {(n^{3})}^{2})$ .

$\frac{9 (n^{4} (m^{2} {(n^{3})}^{2}))}{16 m^{6} \cdot 9^{2}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $9$ out of $16 m^{6} \cdot 9^{2}$ .

$\frac{9 (n^{4} (m^{2} {(n^{3})}^{2}))}{9 (16 m^{6} \cdot 9)}$

Step 9.2.2

Cancel the common factor.

$\frac{9 (n^{4} (m^{2} {(n^{3})}^{2}))}{9 (16 m^{6} \cdot 9)}$

Step 9.2.3

Rewrite the expression.

$\frac{n^{4} (m^{2} {(n^{3})}^{2})}{16 m^{6} \cdot 9}$

Step 10

Cancel the common factor of $m^{2}$ and $m^{6}$ .

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Step 10.1

Factor $m^{2}$ out of $n^{4} (m^{2} {(n^{3})}^{2})$ .

$\frac{m^{2} (n^{4} ({(n^{3})}^{2}))}{16 m^{6} \cdot 9}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $m^{2}$ out of $16 m^{6} \cdot 9$ .

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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

Step 11

Simplify the expression.

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Step 11.1

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

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Step 11.1.1

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

$\frac{n^{4} n^{3 \cdot 2}}{16 m^{4} \cdot 9}$

Step 11.1.2

Multiply $3$ by $2$ .

$\frac{n^{4} n^{6}}{16 m^{4} \cdot 9}$

Step 11.2

Multiply $9$ by $16$ .

$\frac{n^{4} n^{6}}{144 m^{4}}$

Step 12

Multiply $n^{4}$ by $n^{6}$ by adding the exponents.

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Step 12.1

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

$\frac{n^{4 + 6}}{144 m^{4}}$

Step 12.2

Add $4$ and $6$ .

$\frac{n^{10}}{144 m^{4}}$