Simplify (x^2y^4)^(1/2) square root of x^4

${(x^{2} y^{4})}^{\frac{1}{2}} \sqrt{x^{4}}$

Step 1

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

${(x^{2})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}} \sqrt{x^{4}}$

Step 2

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

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

$x^{2 (\frac{1}{2})} {(y^{4})}^{\frac{1}{2}} \sqrt{x^{4}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2 (\frac{1}{2})} {(y^{4})}^{\frac{1}{2}} \sqrt{x^{4}}$

Rewrite the expression.

$x^{1} {(y^{4})}^{\frac{1}{2}} \sqrt{x^{4}}$

Step 3

Simplify.

$x {(y^{4})}^{\frac{1}{2}} \sqrt{x^{4}}$

Step 4

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

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

$x y^{4 (\frac{1}{2})} \sqrt{x^{4}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$x y^{2 (2) \frac{1}{2}} \sqrt{x^{4}}$

Cancel the common factor.

$x y^{2 \cdot 2 \frac{1}{2}} \sqrt{x^{4}}$

Rewrite the expression.

$x y^{2} \sqrt{x^{4}}$

Step 5

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x y^{2} \sqrt{{(x^{2})}^{2}}$

Step 6

Pull terms out from under the radical, assuming positive real numbers.

$x y^{2} x^{2}$

Step 7

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$x^{2} x y^{2}$

Multiply $x^{2}$ by $x$ .

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

$x^{2} x^{1} y^{2}$

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

$x^{2 + 1} y^{2}$

Add $2$ and $1$ .

$x^{3} y^{2}$