Basic Math Examples

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

Step 1

Move $a^{\frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 2

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

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

Step 3

Multiply $a^{\frac{5}{6}}$ by $a^{- \frac{2}{3}}$ by adding the exponents.

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Step 3.1

Move $a^{- \frac{2}{3}}$ .

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

Step 3.2

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

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

Step 3.3

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

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

Step 3.4.2

Multiply $3$ by $2$ .

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 2$ by $2$ .

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

Step 3.6.2

Add $- 4$ and $5$ .

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

Step 4

Multiply $b^{2}$ by $b^{- \frac{1}{5}}$ by adding the exponents.

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Step 4.1

Move $b^{- \frac{1}{5}}$ .

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

Step 4.2

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

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

Step 4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 4.4

Combine $2$ and $\frac{5}{5}$ .

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $2$ by $5$ .

${(4 a^{\frac{1}{6}} b^{\frac{- 1 + 10}{5}})}^{- \frac{1}{2}}$

Step 4.6.2

Add $- 1$ and $10$ .

${(4 a^{\frac{1}{6}} b^{\frac{9}{5}})}^{- \frac{1}{2}}$

Step 5

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

$\frac{1}{{(4 a^{\frac{1}{6}} b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6

Simplify the denominator.

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Step 6.1

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

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Step 6.1.1

Apply the product rule to $4 a^{\frac{1}{6}} b^{\frac{9}{5}}$ .

$\frac{1}{{(4 a^{\frac{1}{6}})}^{\frac{1}{2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.1.2

Apply the product rule to $4 a^{\frac{1}{6}}$ .

$\frac{1}{4^{\frac{1}{2}} {(a^{\frac{1}{6}})}^{\frac{1}{2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.2

Rewrite $4$ as $2^{2}$ .

$\frac{1}{{(2^{2})}^{\frac{1}{2}} {(a^{\frac{1}{6}})}^{\frac{1}{2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.3

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

$\frac{1}{2^{2 (\frac{1}{2})} {(a^{\frac{1}{6}})}^{\frac{1}{2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$\frac{1}{2^{2 (\frac{1}{2})} {(a^{\frac{1}{6}})}^{\frac{1}{2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.4.2

Rewrite the expression.

$\frac{1}{2^{1} {(a^{\frac{1}{6}})}^{\frac{1}{2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.5

Evaluate the exponent.

$\frac{1}{2 {(a^{\frac{1}{6}})}^{\frac{1}{2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.6

Multiply the exponents in ${(a^{\frac{1}{6}})}^{\frac{1}{2}}$ .

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Step 6.6.1

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

$\frac{1}{2 a^{\frac{1}{6} \cdot \frac{1}{2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.6.2

Multiply $\frac{1}{6} \cdot \frac{1}{2}$ .

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Step 6.6.2.1

Multiply $\frac{1}{6}$ by $\frac{1}{2}$ .

$\frac{1}{2 a^{\frac{1}{6 \cdot 2}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.6.2.2

Multiply $6$ by $2$ .

$\frac{1}{2 a^{\frac{1}{12}} {(b^{\frac{9}{5}})}^{\frac{1}{2}}}$

Step 6.7

Multiply the exponents in ${(b^{\frac{9}{5}})}^{\frac{1}{2}}$ .

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Step 6.7.1

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

$\frac{1}{2 a^{\frac{1}{12}} b^{\frac{9}{5} \cdot \frac{1}{2}}}$

Step 6.7.2

Multiply $\frac{9}{5} \cdot \frac{1}{2}$ .

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Step 6.7.2.1

Multiply $\frac{9}{5}$ by $\frac{1}{2}$ .

$\frac{1}{2 a^{\frac{1}{12}} b^{\frac{9}{5 \cdot 2}}}$

Step 6.7.2.2

Multiply $5$ by $2$ .

$\frac{1}{2 a^{\frac{1}{12}} b^{\frac{9}{10}}}$