Calculus Examples

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

Step 1

Rewrite $x^{4} y$ as $x^{3} (x y)$ .

Tap for more steps...

Step 1.1

Factor out $x^{3}$ .

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

Step 1.2

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

Tap for more steps...

Step 3.1

Rewrite the expression using the least common index of $15$ .

Tap for more steps...

Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x y^{2}}$ as ${(x y^{2})}^{\frac{1}{5}}$ .

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

Step 3.1.2

Rewrite ${(x y^{2})}^{\frac{1}{5}}$ as ${(x y^{2})}^{\frac{3}{15}}$ .

$x ({(x y^{2})}^{\frac{3}{15}} \sqrt[3]{x y})$

Step 3.1.3

Rewrite ${(x y^{2})}^{\frac{3}{15}}$ as $\sqrt[15]{{(x y^{2})}^{3}}$ .

$x (\sqrt[15]{{(x y^{2})}^{3}} \sqrt[3]{x y})$

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x y}$ as ${(x y)}^{\frac{1}{3}}$ .

$x (\sqrt[15]{{(x y^{2})}^{3}} {(x y)}^{\frac{1}{3}})$

Step 3.1.5

Rewrite ${(x y)}^{\frac{1}{3}}$ as ${(x y)}^{\frac{5}{15}}$ .

$x (\sqrt[15]{{(x y^{2})}^{3}} {(x y)}^{\frac{5}{15}})$

Step 3.1.6

Rewrite ${(x y)}^{\frac{5}{15}}$ as $\sqrt[15]{{(x y)}^{5}}$ .

$x (\sqrt[15]{{(x y^{2})}^{3}} \sqrt[15]{{(x y)}^{5}})$

Step 3.2

Combine using the product rule for radicals.

$x \sqrt[15]{{(x y^{2})}^{3} {(x y)}^{5}}$

Step 3.3

Apply the product rule to $x y^{2}$ .

$x \sqrt[15]{x^{3} {(y^{2})}^{3} {(x y)}^{5}}$

Step 3.4

Apply the product rule to $x y$ .

$x \sqrt[15]{x^{3} {(y^{2})}^{3} x^{5} y^{5}}$

Step 3.5

Multiply $x^{3}$ by $x^{5}$ by adding the exponents.

Tap for more steps...

Step 3.5.1

Move $x^{5}$ .

$x \sqrt[15]{x^{5} x^{3} {(y^{2})}^{3} y^{5}}$

Step 3.5.2

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

$x \sqrt[15]{x^{5 + 3} {(y^{2})}^{3} y^{5}}$

Step 3.5.3

Add $5$ and $3$ .

$x \sqrt[15]{x^{8} {(y^{2})}^{3} y^{5}}$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$x \sqrt[15]{x^{8} y^{2 \cdot 3} y^{5}}$

Step 3.6.2

Multiply $2$ by $3$ .

$x \sqrt[15]{x^{8} y^{6} y^{5}}$

Step 3.7

Multiply $y^{6}$ by $y^{5}$ by adding the exponents.

Tap for more steps...

Step 3.7.1

Move $y^{5}$ .

$x \sqrt[15]{x^{8} (y^{5} y^{6})}$

Step 3.7.2

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

$x \sqrt[15]{x^{8} y^{5 + 6}}$

Step 3.7.3

Add $5$ and $6$ .

$x \sqrt[15]{x^{8} y^{11}}$