Simplify the Radical Expression (16x^4yz^6)^(1/2)

${(16 x^{4} y z^{6})}^{\frac{1}{2}}$

Step 1

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

Apply the product rule to $16 x^{4} y z^{6}$ .

${(16 x^{4} y)}^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Apply the product rule to $16 x^{4} y$ .

${(16 x^{4})}^{\frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Apply the product rule to $16 x^{4}$ .

$16^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Step 2

Rewrite $16$ as $4^{2}$ .

${(4^{2})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Step 3

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

$4^{2 (\frac{1}{2})} {(x^{4})}^{\frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Step 4

Cancel the common factor of $2$ .

Cancel the common factor.

$4^{2 (\frac{1}{2})} {(x^{4})}^{\frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Rewrite the expression.

$4^{1} {(x^{4})}^{\frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Step 5

Evaluate the exponent.

$4 {(x^{4})}^{\frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Step 6

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

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

$4 x^{4 (\frac{1}{2})} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$4 x^{2 (2) \frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Cancel the common factor.

$4 x^{2 \cdot 2 \frac{1}{2}} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Rewrite the expression.

$4 x^{2} y^{\frac{1}{2}} {(z^{6})}^{\frac{1}{2}}$

Step 7

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

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

$4 x^{2} y^{\frac{1}{2}} z^{6 (\frac{1}{2})}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$4 x^{2} y^{\frac{1}{2}} z^{2 (3) \frac{1}{2}}$

Cancel the common factor.

$4 x^{2} y^{\frac{1}{2}} z^{2 \cdot 3 \frac{1}{2}}$

Rewrite the expression.

$4 x^{2} y^{\frac{1}{2}} z^{3}$

Step 8

Reorder factors in $4 x^{2} y^{\frac{1}{2}} z^{3}$ .

$4 x^{2} z^{3} y^{\frac{1}{2}}$

Simplify the Radical Expression (16x^4yz^6)^(1/2)

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