Simplify (4a^2b^(-3/2))^(1/2)

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

Step 1

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

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

Step 2

Multiply $4 a^{2} \frac{1}{b^{\frac{3}{2}}}$ .

Combine $4$ and $\frac{1}{b^{\frac{3}{2}}}$ .

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

Combine $a^{2}$ and $\frac{4}{b^{\frac{3}{2}}}$ .

${(\frac{a^{2} \cdot 4}{b^{\frac{3}{2}}})}^{\frac{1}{2}}$

Step 3

Move $4$ to the left of $a^{2}$ .

${(\frac{4 \cdot a^{2}}{b^{\frac{3}{2}}})}^{\frac{1}{2}}$

Step 4

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

Apply the product rule to $\frac{4 a^{2}}{b^{\frac{3}{2}}}$ .

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

Apply the product rule to $4 a^{2}$ .

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

Step 5

Simplify the numerator.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 6

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

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

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

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

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

Multiply $2$ by $2$ .

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

Simplify (4a^2b^(-3/2))^(1/2)

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