Simplify (ab^4*b^2x^4)^(1/2)

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

Step 1

Multiply $b^{4}$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

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

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

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

Add $2$ and $4$ .

${(a b^{6} \cdot x^{4})}^{\frac{1}{2}}$

Step 2

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

Apply the product rule to $a b^{6} x^{4}$ .

${(a b^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}$

Apply the product rule to $a b^{6}$ .

$a^{\frac{1}{2}} {(b^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}$

Step 3

Multiply the exponents in ${(b^{6})}^{\frac{1}{2}}$ .

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

$a^{\frac{1}{2}} b^{6 (\frac{1}{2})} {(x^{4})}^{\frac{1}{2}}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Reorder factors in $a^{\frac{1}{2}} b^{3} x^{2}$ .

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