Simplify the Radical Expression (x^3y^7z^4)^(1/2)

${(x^{3} y^{7} z^{4})}^{\frac{1}{2}}$

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

Apply the product rule to $x^{3} y^{7} z^{4}$ .

${(x^{3} y^{7})}^{\frac{1}{2}} {(z^{4})}^{\frac{1}{2}}$

Apply the product rule to $x^{3} y^{7}$ .

${(x^{3})}^{\frac{1}{2}} {(y^{7})}^{\frac{1}{2}} {(z^{4})}^{\frac{1}{2}}$

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

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

$x^{3 (\frac{1}{2})} {(y^{7})}^{\frac{1}{2}} {(z^{4})}^{\frac{1}{2}}$

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

$x^{\frac{3}{2}} {(y^{7})}^{\frac{1}{2}} {(z^{4})}^{\frac{1}{2}}$

Multiply the exponents in ${(y^{7})}^{\frac{1}{2}}$ .

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

$x^{\frac{3}{2}} y^{7 (\frac{1}{2})} {(z^{4})}^{\frac{1}{2}}$

Combine $7$ and $\frac{1}{2}$ .

$x^{\frac{3}{2}} y^{\frac{7}{2}} {(z^{4})}^{\frac{1}{2}}$

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

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

$x^{\frac{3}{2}} y^{\frac{7}{2}} z^{4 (\frac{1}{2})}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$x^{\frac{3}{2}} y^{\frac{7}{2}} z^{2 (2) \frac{1}{2}}$

Cancel the common factor.

$x^{\frac{3}{2}} y^{\frac{7}{2}} z^{2 \cdot 2 \frac{1}{2}}$

Rewrite the expression.

$x^{\frac{3}{2}} y^{\frac{7}{2}} z^{2}$

Reorder factors in $x^{\frac{3}{2}} y^{\frac{7}{2}} z^{2}$ .

$z^{2} x^{\frac{3}{2}} y^{\frac{7}{2}}$