Simplify the Radical Expression (4r^2s^3t^4)^(2/5)

${(4 r^{2} s^{3} t^{4})}^{\frac{2}{5}}$

Step 1

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

Apply the product rule to $4 r^{2} s^{3} t^{4}$ .

${(4 r^{2} s^{3})}^{\frac{2}{5}} {(t^{4})}^{\frac{2}{5}}$

Apply the product rule to $4 r^{2} s^{3}$ .

${(4 r^{2})}^{\frac{2}{5}} {(s^{3})}^{\frac{2}{5}} {(t^{4})}^{\frac{2}{5}}$

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

$4^{\frac{2}{5}} {(r^{2})}^{\frac{2}{5}} {(s^{3})}^{\frac{2}{5}} {(t^{4})}^{\frac{2}{5}}$

Step 2

Multiply the exponents in ${(r^{2})}^{\frac{2}{5}}$ .

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

$4^{\frac{2}{5}} r^{2 (\frac{2}{5})} {(s^{3})}^{\frac{2}{5}} {(t^{4})}^{\frac{2}{5}}$

Multiply $2 (\frac{2}{5})$ .

Combine $2$ and $\frac{2}{5}$ .

$4^{\frac{2}{5}} r^{\frac{2 \cdot 2}{5}} {(s^{3})}^{\frac{2}{5}} {(t^{4})}^{\frac{2}{5}}$

Multiply $2$ by $2$ .

$4^{\frac{2}{5}} r^{\frac{4}{5}} {(s^{3})}^{\frac{2}{5}} {(t^{4})}^{\frac{2}{5}}$

Step 3

Multiply the exponents in ${(s^{3})}^{\frac{2}{5}}$ .

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

$4^{\frac{2}{5}} r^{\frac{4}{5}} s^{3 (\frac{2}{5})} {(t^{4})}^{\frac{2}{5}}$

Multiply $3 (\frac{2}{5})$ .

Combine $3$ and $\frac{2}{5}$ .

$4^{\frac{2}{5}} r^{\frac{4}{5}} s^{\frac{3 \cdot 2}{5}} {(t^{4})}^{\frac{2}{5}}$

Multiply $3$ by $2$ .

$4^{\frac{2}{5}} r^{\frac{4}{5}} s^{\frac{6}{5}} {(t^{4})}^{\frac{2}{5}}$

Step 4

Multiply the exponents in ${(t^{4})}^{\frac{2}{5}}$ .

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

$4^{\frac{2}{5}} r^{\frac{4}{5}} s^{\frac{6}{5}} t^{4 (\frac{2}{5})}$

Multiply $4 (\frac{2}{5})$ .

Combine $4$ and $\frac{2}{5}$ .

$4^{\frac{2}{5}} r^{\frac{4}{5}} s^{\frac{6}{5}} t^{\frac{4 \cdot 2}{5}}$

Multiply $4$ by $2$ .

$4^{\frac{2}{5}} r^{\frac{4}{5}} s^{\frac{6}{5}} t^{\frac{8}{5}}$