Basic Math Examples

${(\frac{3 a^{- 1} b^{4}}{5 b^{- 4} a^{- 6}})}^{- 2}$

Step 1

Move $a^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{3 a^{- 1} b^{4} a^{6}}{5 b^{- 4}})}^{- 2}$

Step 2

Move $b^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{3 a^{- 1} b^{4} a^{6} b^{4}}{5})}^{- 2}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $a^{- 1}$ by $a^{6}$ by adding the exponents.

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Step 3.1.1

Move $a^{6}$ .

${(\frac{3 (a^{6} a^{- 1}) b^{4} b^{4}}{5})}^{- 2}$

Step 3.1.2

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

${(\frac{3 a^{6 - 1} b^{4} b^{4}}{5})}^{- 2}$

Step 3.1.3

Subtract $1$ from $6$ .

${(\frac{3 a^{5} b^{4} b^{4}}{5})}^{- 2}$

Step 3.2

Multiply $b^{4}$ by $b^{4}$ by adding the exponents.

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Step 3.2.1

Move $b^{4}$ .

${(\frac{3 a^{5} (b^{4} b^{4})}{5})}^{- 2}$

Step 3.2.2

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

${(\frac{3 a^{5} b^{4 + 4}}{5})}^{- 2}$

Step 3.2.3

Add $4$ and $4$ .

${(\frac{3 a^{5} b^{8}}{5})}^{- 2}$

Step 4

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

${(\frac{5}{3 a^{5} b^{8}})}^{2}$

Step 5

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

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Step 5.1

Apply the product rule to $\frac{5}{3 a^{5} b^{8}}$ .

$\frac{5^{2}}{{(3 a^{5} b^{8})}^{2}}$

Step 5.2

Apply the product rule to $3 a^{5} b^{8}$ .

$\frac{5^{2}}{{(3 a^{5})}^{2} {(b^{8})}^{2}}$

Step 5.3

Apply the product rule to $3 a^{5}$ .

$\frac{5^{2}}{3^{2} {(a^{5})}^{2} {(b^{8})}^{2}}$

Step 6

Raise $5$ to the power of $2$ .

$\frac{25}{3^{2} {(a^{5})}^{2} {(b^{8})}^{2}}$

Step 7

Simplify the denominator.

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Step 7.1

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

$\frac{25}{9 {(a^{5})}^{2} {(b^{8})}^{2}}$

Step 7.2

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

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Step 7.2.1

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

$\frac{25}{9 a^{5 \cdot 2} {(b^{8})}^{2}}$

Step 7.2.2

Multiply $5$ by $2$ .

$\frac{25}{9 a^{10} {(b^{8})}^{2}}$

Step 7.3

Multiply the exponents in ${(b^{8})}^{2}$ .

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Step 7.3.1

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

$\frac{25}{9 a^{10} b^{8 \cdot 2}}$

Step 7.3.2

Multiply $8$ by $2$ .

$\frac{25}{9 a^{10} b^{16}}$