Simplify (a^(1/2)b)^(1/2)(ab^(1/2))

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

Step 1

Apply the product rule to $a^{\frac{1}{2}} b$ .

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

Step 2

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Step 3

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

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

Step 4

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

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

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

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

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

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

Multiply $2$ by $2$ .

$a^{\frac{1}{4}} b a$

Step 5

Multiply $a^{\frac{1}{4}}$ by $a$ by adding the exponents.

Move $a$ .

$a \cdot a^{\frac{1}{4}} b$

Multiply $a$ by $a^{\frac{1}{4}}$ .

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

$a^{1} a^{\frac{1}{4}} b$

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

$a^{1 + \frac{1}{4}} b$

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

$a^{\frac{4}{4} + \frac{1}{4}} b$

Combine the numerators over the common denominator.

$a^{\frac{4 + 1}{4}} b$

Add $4$ and $1$ .

$a^{\frac{5}{4}} b$

Step 6

Reorder factors in $a^{\frac{5}{4}} b$ .

$b a^{\frac{5}{4}}$

Simplify (a^(1/2)b)^(1/2)(ab^(1/2))

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