Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Combine $5$ and $\frac{1}{2}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Apply the product rule to $a b$ .

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

Step 6

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

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Step 6.1

Move $a^{\frac{1}{2}}$ .

$a^{\frac{1}{2}} a^{\frac{5}{2}} b^{\frac{3}{2}} b^{\frac{1}{2}}$

Step 6.2

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

$a^{\frac{1}{2} + \frac{5}{2}} b^{\frac{3}{2}} b^{\frac{1}{2}}$

Step 6.3

Combine the numerators over the common denominator.

$a^{\frac{1 + 5}{2}} b^{\frac{3}{2}} b^{\frac{1}{2}}$

Step 6.4

Add $1$ and $5$ .

$a^{\frac{6}{2}} b^{\frac{3}{2}} b^{\frac{1}{2}}$

Step 6.5

Divide $6$ by $2$ .

$a^{3} b^{\frac{3}{2}} b^{\frac{1}{2}}$

Step 7

Multiply $b^{\frac{3}{2}}$ by $b^{\frac{1}{2}}$ by adding the exponents.

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Step 7.1

Move $b^{\frac{1}{2}}$ .

$a^{3} (b^{\frac{1}{2}} b^{\frac{3}{2}})$

Step 7.2

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

$a^{3} b^{\frac{1}{2} + \frac{3}{2}}$

Step 7.3

Combine the numerators over the common denominator.

$a^{3} b^{\frac{1 + 3}{2}}$

Step 7.4

Add $1$ and $3$ .

$a^{3} b^{\frac{4}{2}}$

Step 7.5

Divide $4$ by $2$ .

$a^{3} b^{2}$