Simplify (m^(1/4)n^(1/2))^2(m^2n^3)^(1/2)

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

Step 1

Apply the product rule to $m^{\frac{1}{4}} n^{\frac{1}{2}}$ .

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

Step 2

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

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

$m^{\frac{1}{4} \cdot 2} {(n^{\frac{1}{2}})}^{2} {(m^{2} n^{3})}^{\frac{1}{2}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$m^{\frac{1}{2 (2)} \cdot 2} {(n^{\frac{1}{2}})}^{2} {(m^{2} n^{3})}^{\frac{1}{2}}$

Cancel the common factor.

$m^{\frac{1}{2 \cdot 2} \cdot 2} {(n^{\frac{1}{2}})}^{2} {(m^{2} n^{3})}^{\frac{1}{2}}$

Rewrite the expression.

$m^{\frac{1}{2}} {(n^{\frac{1}{2}})}^{2} {(m^{2} n^{3})}^{\frac{1}{2}}$

Step 3

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

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

$m^{\frac{1}{2}} n^{\frac{1}{2} \cdot 2} {(m^{2} n^{3})}^{\frac{1}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$m^{\frac{1}{2}} n^{\frac{1}{2} \cdot 2} {(m^{2} n^{3})}^{\frac{1}{2}}$

Rewrite the expression.

$m^{\frac{1}{2}} n^{1} {(m^{2} n^{3})}^{\frac{1}{2}}$

Step 4

Simplify.

$m^{\frac{1}{2}} n {(m^{2} n^{3})}^{\frac{1}{2}}$

Step 5

Apply the product rule to $m^{2} n^{3}$ .

$m^{\frac{1}{2}} n ({(m^{2})}^{\frac{1}{2}} {(n^{3})}^{\frac{1}{2}})$

Step 6

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

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

$m^{\frac{1}{2}} n (m^{2 (\frac{1}{2})} {(n^{3})}^{\frac{1}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$m^{\frac{1}{2}} n (m^{2 (\frac{1}{2})} {(n^{3})}^{\frac{1}{2}})$

Rewrite the expression.

$m^{\frac{1}{2}} n (m^{1} {(n^{3})}^{\frac{1}{2}})$

Step 7

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

Move $m^{1}$ .

$m^{1} m^{\frac{1}{2}} n {(n^{3})}^{\frac{1}{2}}$

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

$m^{1 + \frac{1}{2}} n {(n^{3})}^{\frac{1}{2}}$

Write $1$ as a fraction with a common denominator.

$m^{\frac{2}{2} + \frac{1}{2}} n {(n^{3})}^{\frac{1}{2}}$

Combine the numerators over the common denominator.

$m^{\frac{2 + 1}{2}} n {(n^{3})}^{\frac{1}{2}}$

Add $2$ and $1$ .

$m^{\frac{3}{2}} n {(n^{3})}^{\frac{1}{2}}$

Step 8

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

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

$m^{\frac{3}{2}} n \cdot n^{3 (\frac{1}{2})}$

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

$m^{\frac{3}{2}} n \cdot n^{\frac{3}{2}}$

Step 9

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

Move $n^{\frac{3}{2}}$ .

$m^{\frac{3}{2}} (n^{\frac{3}{2}} n)$

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

Raise $n$ to the power of $1$ .

$m^{\frac{3}{2}} (n^{\frac{3}{2}} n^{1})$

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

$m^{\frac{3}{2}} n^{\frac{3}{2} + 1}$

Write $1$ as a fraction with a common denominator.

$m^{\frac{3}{2}} n^{\frac{3}{2} + \frac{2}{2}}$

Combine the numerators over the common denominator.

$m^{\frac{3}{2}} n^{\frac{3 + 2}{2}}$

Add $3$ and $2$ .

$m^{\frac{3}{2}} n^{\frac{5}{2}}$

Simplify (m^(1/4)n^(1/2))^2(m^2n^3)^(1/2)

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