Simplify the Radical Expression (5x^4y^3)^(2/9)

${(5 x^{4} y^{3})}^{\frac{2}{9}}$

Step 1

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

Apply the product rule to $5 x^{4} y^{3}$ .

${(5 x^{4})}^{\frac{2}{9}} {(y^{3})}^{\frac{2}{9}}$

Apply the product rule to $5 x^{4}$ .

$5^{\frac{2}{9}} {(x^{4})}^{\frac{2}{9}} {(y^{3})}^{\frac{2}{9}}$

Step 2

Multiply the exponents in ${(x^{4})}^{\frac{2}{9}}$ .

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

$5^{\frac{2}{9}} x^{4 (\frac{2}{9})} {(y^{3})}^{\frac{2}{9}}$

Multiply $4 (\frac{2}{9})$ .

Combine $4$ and $\frac{2}{9}$ .

$5^{\frac{2}{9}} x^{\frac{4 \cdot 2}{9}} {(y^{3})}^{\frac{2}{9}}$

Multiply $4$ by $2$ .

$5^{\frac{2}{9}} x^{\frac{8}{9}} {(y^{3})}^{\frac{2}{9}}$

Step 3

Multiply the exponents in ${(y^{3})}^{\frac{2}{9}}$ .

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

$5^{\frac{2}{9}} x^{\frac{8}{9}} y^{3 (\frac{2}{9})}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$5^{\frac{2}{9}} x^{\frac{8}{9}} y^{3 \frac{2}{3 (3)}}$

Cancel the common factor.

$5^{\frac{2}{9}} x^{\frac{8}{9}} y^{3 \frac{2}{3 \cdot 3}}$

Rewrite the expression.

$5^{\frac{2}{9}} x^{\frac{8}{9}} y^{\frac{2}{3}}$