Calculus Examples

$\frac{\sqrt{a \sqrt{b}}}{\sqrt[3]{a b}}$

Step 1

Multiply $\frac{\sqrt{a \sqrt{b}}}{\sqrt[3]{a b}}$ by $\frac{{\sqrt[3]{a b}}^{2}}{{\sqrt[3]{a b}}^{2}}$ .

$\frac{\sqrt{a \sqrt{b}}}{\sqrt[3]{a b}} \cdot \frac{{\sqrt[3]{a b}}^{2}}{{\sqrt[3]{a b}}^{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt{a \sqrt{b}}}{\sqrt[3]{a b}}$ by $\frac{{\sqrt[3]{a b}}^{2}}{{\sqrt[3]{a b}}^{2}}$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{\sqrt[3]{a b} {\sqrt[3]{a b}}^{2}}$

Step 2.2

Raise $\sqrt[3]{a b}$ to the power of $1$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{{\sqrt[3]{a b}}^{1} {\sqrt[3]{a b}}^{2}}$

Step 2.3

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

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{{\sqrt[3]{a b}}^{1 + 2}}$

Step 2.4

Add $1$ and $2$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{{\sqrt[3]{a b}}^{3}}$

Step 2.5

Rewrite ${\sqrt[3]{a b}}^{3}$ as $a b$ .

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Step 2.5.1

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

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{{({(a b)}^{\frac{1}{3}})}^{3}}$

Step 2.5.2

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

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{{(a b)}^{\frac{1}{3} \cdot 3}}$

Step 2.5.3

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

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{{(a b)}^{\frac{3}{3}}}$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{{(a b)}^{\frac{3}{3}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{{(a b)}^{1}}$

Step 2.5.5

Simplify.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[3]{a b}}^{2}}{a b}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{a b}}^{2}$ as $\sqrt[3]{{(a b)}^{2}}$ .

$\frac{\sqrt{a \sqrt{b}} \sqrt[3]{{(a b)}^{2}}}{a b}$

Step 3.2

Apply the product rule to $a b$ .

$\frac{\sqrt{a \sqrt{b}} \sqrt[3]{a^{2} b^{2}}}{a b}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $6$ .

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Step 4.1.1

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

$\frac{{(a \sqrt{b})}^{\frac{1}{2}} \sqrt[3]{a^{2} b^{2}}}{a b}$

Step 4.1.2

Rewrite ${(a \sqrt{b})}^{\frac{1}{2}}$ as ${(a \sqrt{b})}^{\frac{3}{6}}$ .

$\frac{{(a \sqrt{b})}^{\frac{3}{6}} \sqrt[3]{a^{2} b^{2}}}{a b}$

Step 4.1.3

Rewrite ${(a \sqrt{b})}^{\frac{3}{6}}$ as $\sqrt[6]{{(a \sqrt{b})}^{3}}$ .

$\frac{\sqrt[6]{{(a \sqrt{b})}^{3}} \sqrt[3]{a^{2} b^{2}}}{a b}$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{a^{2} b^{2}}$ as ${(a^{2} b^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[6]{{(a \sqrt{b})}^{3}} {(a^{2} b^{2})}^{\frac{1}{3}}}{a b}$

Step 4.1.5

Rewrite ${(a^{2} b^{2})}^{\frac{1}{3}}$ as ${(a^{2} b^{2})}^{\frac{2}{6}}$ .

$\frac{\sqrt[6]{{(a \sqrt{b})}^{3}} {(a^{2} b^{2})}^{\frac{2}{6}}}{a b}$

Step 4.1.6

Rewrite ${(a^{2} b^{2})}^{\frac{2}{6}}$ as $\sqrt[6]{{(a^{2} b^{2})}^{2}}$ .

$\frac{\sqrt[6]{{(a \sqrt{b})}^{3}} \sqrt[6]{{(a^{2} b^{2})}^{2}}}{a b}$

Step 4.2

Combine using the product rule for radicals.

$\frac{\sqrt[6]{{(a \sqrt{b})}^{3} {(a^{2} b^{2})}^{2}}}{a b}$

Step 4.3

Apply the product rule to $a \sqrt{b}$ .

$\frac{\sqrt[6]{a^{3} {\sqrt{b}}^{3} {(a^{2} b^{2})}^{2}}}{a b}$

Step 4.4

Apply the product rule to $a^{2} b^{2}$ .

$\frac{\sqrt[6]{a^{3} {\sqrt{b}}^{3} {(a^{2})}^{2} {(b^{2})}^{2}}}{a b}$

Step 4.5

Rewrite ${\sqrt{b}}^{3}$ as $\sqrt{b^{3}}$ .

$\frac{\sqrt[6]{a^{3} \sqrt{b^{3}} {(a^{2})}^{2} {(b^{2})}^{2}}}{a b}$

Step 4.6

Factor out $b^{2}$ .

$\frac{\sqrt[6]{a^{3} \sqrt{b^{2} b} {(a^{2})}^{2} {(b^{2})}^{2}}}{a b}$

Step 4.7

Pull terms out from under the radical.

$\frac{\sqrt[6]{a^{3} (b \sqrt{b}) {(a^{2})}^{2} {(b^{2})}^{2}}}{a b}$

Step 4.8

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

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Step 4.8.1

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

$\frac{\sqrt[6]{a^{3} (b \sqrt{b}) a^{2 \cdot 2} {(b^{2})}^{2}}}{a b}$

Step 4.8.2

Multiply $2$ by $2$ .

$\frac{\sqrt[6]{a^{3} (b \sqrt{b}) a^{4} {(b^{2})}^{2}}}{a b}$

Step 4.9

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

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Step 4.9.1

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

$\frac{\sqrt[6]{a^{3} (b \sqrt{b}) a^{4} b^{2 \cdot 2}}}{a b}$

Step 4.9.2

Multiply $2$ by $2$ .

$\frac{\sqrt[6]{a^{3} (b \sqrt{b}) a^{4} b^{4}}}{a b}$

$\frac{\sqrt[6]{a^{3} b \sqrt{b} a^{4} b^{4}}}{a b}$

Step 4.10

Combine exponents.

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Step 4.10.1

Multiply $a^{3}$ by $a^{4}$ by adding the exponents.

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Step 4.10.1.1

Move $a^{4}$ .

$\frac{\sqrt[6]{a^{4} a^{3} b \sqrt{b} b^{4}}}{a b}$

Step 4.10.1.2

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

$\frac{\sqrt[6]{a^{4 + 3} b \sqrt{b} b^{4}}}{a b}$

Step 4.10.1.3

Add $4$ and $3$ .

$\frac{\sqrt[6]{a^{7} b \sqrt{b} b^{4}}}{a b}$

Step 4.10.2

Multiply $b$ by $b^{4}$ by adding the exponents.

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Step 4.10.2.1

Move $b^{4}$ .

$\frac{\sqrt[6]{a^{7} (b^{4} b) \sqrt{b}}}{a b}$

Step 4.10.2.2

Multiply $b^{4}$ by $b$ .

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Step 4.10.2.2.1

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

$\frac{\sqrt[6]{a^{7} (b^{4} b^{1}) \sqrt{b}}}{a b}$

Step 4.10.2.2.2

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

$\frac{\sqrt[6]{a^{7} b^{4 + 1} \sqrt{b}}}{a b}$

Step 4.10.2.3

Add $4$ and $1$ .

$\frac{\sqrt[6]{a^{7} b^{5} \sqrt{b}}}{a b}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $a^{7} b^{5} \sqrt{b}$ as $a^{6} (a b^{5} \sqrt{b})$ .

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Step 5.1.1

Factor out $a^{6}$ .

$\frac{\sqrt[6]{a^{6} a b^{5} \sqrt{b}}}{a b}$

Step 5.1.2

Add parentheses.

$\frac{\sqrt[6]{a^{6} a (b^{5} \sqrt{b})}}{a b}$

Step 5.1.3

Add parentheses.

$\frac{\sqrt[6]{a^{6} (a b^{5} \sqrt{b})}}{a b}$

Step 5.2

Pull terms out from under the radical.

$\frac{a \sqrt[6]{a b^{5} \sqrt{b}}}{a b}$

Step 6

Cancel the common factor of $a$ .

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Step 6.1

Cancel the common factor.

$\frac{a \sqrt[6]{a b^{5} \sqrt{b}}}{a b}$

Step 6.2

Rewrite the expression.

$\frac{\sqrt[6]{a b^{5} \sqrt{b}}}{b}$