Simplify ((64m^(1/2)r^3)/(m^(5/2)r^(1/4)))^(1/2)

${(\frac{64 m^{\frac{1}{2}} r^{3}}{m^{\frac{5}{2}} r^{\frac{1}{4}}})}^{\frac{1}{2}}$

Step 1

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

${(\frac{64 r^{3}}{m^{\frac{5}{2}} r^{\frac{1}{4}} m^{- \frac{1}{2}}})}^{\frac{1}{2}}$

Step 2

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

Move $m^{- \frac{1}{2}}$ .

${(\frac{64 r^{3}}{m^{- \frac{1}{2}} m^{\frac{5}{2}} r^{\frac{1}{4}}})}^{\frac{1}{2}}$

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

${(\frac{64 r^{3}}{m^{- \frac{1}{2} + \frac{5}{2}} r^{\frac{1}{4}}})}^{\frac{1}{2}}$

Combine the numerators over the common denominator.

${(\frac{64 r^{3}}{m^{\frac{- 1 + 5}{2}} r^{\frac{1}{4}}})}^{\frac{1}{2}}$

Add $- 1$ and $5$ .

${(\frac{64 r^{3}}{m^{\frac{4}{2}} r^{\frac{1}{4}}})}^{\frac{1}{2}}$

Divide $4$ by $2$ .

${(\frac{64 r^{3}}{m^{2} r^{\frac{1}{4}}})}^{\frac{1}{2}}$

Step 3

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

${(\frac{64 r^{3} r^{- \frac{1}{4}}}{m^{2}})}^{\frac{1}{2}}$

Step 4

Multiply $r^{3}$ by $r^{- \frac{1}{4}}$ by adding the exponents.

Move $r^{- \frac{1}{4}}$ .

${(\frac{64 (r^{- \frac{1}{4}} r^{3})}{m^{2}})}^{\frac{1}{2}}$

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

${(\frac{64 r^{- \frac{1}{4} + 3}}{m^{2}})}^{\frac{1}{2}}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(\frac{64 r^{- \frac{1}{4} + 3 \cdot \frac{4}{4}}}{m^{2}})}^{\frac{1}{2}}$

Combine $3$ and $\frac{4}{4}$ .

${(\frac{64 r^{- \frac{1}{4} + \frac{3 \cdot 4}{4}}}{m^{2}})}^{\frac{1}{2}}$

Combine the numerators over the common denominator.

${(\frac{64 r^{\frac{- 1 + 3 \cdot 4}{4}}}{m^{2}})}^{\frac{1}{2}}$

Simplify the numerator.

Multiply $3$ by $4$ .

${(\frac{64 r^{\frac{- 1 + 12}{4}}}{m^{2}})}^{\frac{1}{2}}$

Add $- 1$ and $12$ .

${(\frac{64 r^{\frac{11}{4}}}{m^{2}})}^{\frac{1}{2}}$

Step 5

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

Apply the product rule to $\frac{64 r^{\frac{11}{4}}}{m^{2}}$ .

$\frac{{(64 r^{\frac{11}{4}})}^{\frac{1}{2}}}{{(m^{2})}^{\frac{1}{2}}}$

Apply the product rule to $64 r^{\frac{11}{4}}$ .

$\frac{64^{\frac{1}{2}} {(r^{\frac{11}{4}})}^{\frac{1}{2}}}{{(m^{2})}^{\frac{1}{2}}}$

Step 6

Simplify the numerator.

Rewrite $64$ as $8^{2}$ .

$\frac{{(8^{2})}^{\frac{1}{2}} {(r^{\frac{11}{4}})}^{\frac{1}{2}}}{{(m^{2})}^{\frac{1}{2}}}$

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

$\frac{8^{2 (\frac{1}{2})} {(r^{\frac{11}{4}})}^{\frac{1}{2}}}{{(m^{2})}^{\frac{1}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{8^{2 (\frac{1}{2})} {(r^{\frac{11}{4}})}^{\frac{1}{2}}}{{(m^{2})}^{\frac{1}{2}}}$

Rewrite the expression.

$\frac{8^{1} {(r^{\frac{11}{4}})}^{\frac{1}{2}}}{{(m^{2})}^{\frac{1}{2}}}$

Evaluate the exponent.

$\frac{8 {(r^{\frac{11}{4}})}^{\frac{1}{2}}}{{(m^{2})}^{\frac{1}{2}}}$

Multiply the exponents in ${(r^{\frac{11}{4}})}^{\frac{1}{2}}$ .

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

$\frac{8 r^{\frac{11}{4} \cdot \frac{1}{2}}}{{(m^{2})}^{\frac{1}{2}}}$

Multiply $\frac{11}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{11}{4}$ by $\frac{1}{2}$ .

$\frac{8 r^{\frac{11}{4 \cdot 2}}}{{(m^{2})}^{\frac{1}{2}}}$

Multiply $4$ by $2$ .

$\frac{8 r^{\frac{11}{8}}}{{(m^{2})}^{\frac{1}{2}}}$

Step 7

Simplify the denominator.

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

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

$\frac{8 r^{\frac{11}{8}}}{m^{2 (\frac{1}{2})}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{8 r^{\frac{11}{8}}}{m^{2 (\frac{1}{2})}}$

Rewrite the expression.

$\frac{8 r^{\frac{11}{8}}}{m^{1}}$

Simplify.

$\frac{8 r^{\frac{11}{8}}}{m}$

Simplify ((64m^(1/2)r^3)/(m^(5/2)r^(1/4)))^(1/2)

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