Simplify fifth root of 125x^4y^3* fifth root of 25xy^2

$\sqrt[5]{125 x^{4} y^{3}} \cdot \sqrt[5]{25 x y^{2}}$

Step 1

Multiply $\sqrt[5]{125 x^{4} y^{3}} \sqrt[5]{25 x y^{2}}$ .

Combine using the product rule for radicals.

$\sqrt[5]{125 x^{4} y^{3} (25 x y^{2})}$

Multiply $25$ by $125$ .

$\sqrt[5]{3125 x^{4} y^{3} (x y^{2})}$

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

$\sqrt[5]{3125 (x^{1} x^{4}) y^{3} y^{2}}$

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

$\sqrt[5]{3125 x^{1 + 4} y^{3} y^{2}}$

Add $1$ and $4$ .

$\sqrt[5]{3125 x^{5} y^{3} y^{2}}$

Multiply $y^{3}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$\sqrt[5]{3125 x^{5} (y^{2} y^{3})}$

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

$\sqrt[5]{3125 x^{5} y^{2 + 3}}$

Add $2$ and $3$ .

$\sqrt[5]{3125 x^{5} y^{5}}$

Step 2

Rewrite $3125 x^{5} y^{5}$ as ${(5 x y)}^{5}$ .

$\sqrt[5]{{(5 x y)}^{5}}$

Step 3

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

$5 x y$