Algebra Examples

$\frac{\sqrt{a \sqrt{b}}}{\sqrt[9]{a b}}$

Step 1

Multiply $\frac{\sqrt{a \sqrt{b}}}{\sqrt[9]{a b}}$ by $\frac{{\sqrt[9]{a b}}^{8}}{{\sqrt[9]{a b}}^{8}}$ .

$\frac{\sqrt{a \sqrt{b}}}{\sqrt[9]{a b}} \cdot \frac{{\sqrt[9]{a b}}^{8}}{{\sqrt[9]{a b}}^{8}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt{a \sqrt{b}}}{\sqrt[9]{a b}}$ by $\frac{{\sqrt[9]{a b}}^{8}}{{\sqrt[9]{a b}}^{8}}$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{\sqrt[9]{a b} {\sqrt[9]{a b}}^{8}}$

Step 2.2

Raise $\sqrt[9]{a b}$ to the power of $1$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{{\sqrt[9]{a b}}^{1} {\sqrt[9]{a b}}^{8}}$

Step 2.3

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

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{{\sqrt[9]{a b}}^{1 + 8}}$

Step 2.4

Add $1$ and $8$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{{\sqrt[9]{a b}}^{9}}$

Step 2.5

Rewrite ${\sqrt[9]{a b}}^{9}$ as $a b$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{a b}$ as ${(a b)}^{\frac{1}{9}}$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{{({(a b)}^{\frac{1}{9}})}^{9}}$

Step 2.5.2

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

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{{(a b)}^{\frac{1}{9} \cdot 9}}$

Step 2.5.3

Combine $\frac{1}{9}$ and $9$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{{(a b)}^{\frac{9}{9}}}$

Step 2.5.4

Cancel the common factor of $9$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{{(a b)}^{\frac{9}{9}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{{(a b)}^{1}}$

Step 2.5.5

Simplify.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[9]{a b}}^{8}}{a b}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[9]{a b}}^{8}$ as $\sqrt[9]{{(a b)}^{8}}$ .

$\frac{\sqrt{a \sqrt{b}} \sqrt[9]{{(a b)}^{8}}}{a b}$

Step 3.2

Apply the product rule to $a b$ .

$\frac{\sqrt{a \sqrt{b}} \sqrt[9]{a^{8} b^{8}}}{a b}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $18$ .

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Step 4.1.1

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

$\frac{{(a \sqrt{b})}^{\frac{1}{2}} \sqrt[9]{a^{8} b^{8}}}{a b}$

Step 4.1.2

Rewrite ${(a \sqrt{b})}^{\frac{1}{2}}$ as ${(a \sqrt{b})}^{\frac{9}{18}}$ .

$\frac{{(a \sqrt{b})}^{\frac{9}{18}} \sqrt[9]{a^{8} b^{8}}}{a b}$

Step 4.1.3

Rewrite ${(a \sqrt{b})}^{\frac{9}{18}}$ as $\sqrt[18]{{(a \sqrt{b})}^{9}}$ .

$\frac{\sqrt[18]{{(a \sqrt{b})}^{9}} \sqrt[9]{a^{8} b^{8}}}{a b}$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{a^{8} b^{8}}$ as ${(a^{8} b^{8})}^{\frac{1}{9}}$ .

$\frac{\sqrt[18]{{(a \sqrt{b})}^{9}} {(a^{8} b^{8})}^{\frac{1}{9}}}{a b}$

Step 4.1.5

Rewrite ${(a^{8} b^{8})}^{\frac{1}{9}}$ as ${(a^{8} b^{8})}^{\frac{2}{18}}$ .

$\frac{\sqrt[18]{{(a \sqrt{b})}^{9}} {(a^{8} b^{8})}^{\frac{2}{18}}}{a b}$

Step 4.1.6

Rewrite ${(a^{8} b^{8})}^{\frac{2}{18}}$ as $\sqrt[18]{{(a^{8} b^{8})}^{2}}$ .

$\frac{\sqrt[18]{{(a \sqrt{b})}^{9}} \sqrt[18]{{(a^{8} b^{8})}^{2}}}{a b}$

Step 4.2

Combine using the product rule for radicals.

$\frac{\sqrt[18]{{(a \sqrt{b})}^{9} {(a^{8} b^{8})}^{2}}}{a b}$

Step 4.3

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

$\frac{\sqrt[18]{a^{9} {\sqrt{b}}^{9} {(a^{8} b^{8})}^{2}}}{a b}$

Step 4.4

Apply the product rule to $a^{8} b^{8}$ .

$\frac{\sqrt[18]{a^{9} {\sqrt{b}}^{9} {(a^{8})}^{2} {(b^{8})}^{2}}}{a b}$

Step 4.5

Rewrite ${\sqrt{b}}^{9}$ as $\sqrt{b^{9}}$ .

$\frac{\sqrt[18]{a^{9} \sqrt{b^{9}} {(a^{8})}^{2} {(b^{8})}^{2}}}{a b}$

Step 4.6

Rewrite $b^{9}$ as ${(b^{4})}^{2} b$ .

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Step 4.6.1

Factor out $b^{8}$ .

$\frac{\sqrt[18]{a^{9} \sqrt{b^{8} b} {(a^{8})}^{2} {(b^{8})}^{2}}}{a b}$

Step 4.6.2

Rewrite $b^{8}$ as ${(b^{4})}^{2}$ .

$\frac{\sqrt[18]{a^{9} \sqrt{{(b^{4})}^{2} b} {(a^{8})}^{2} {(b^{8})}^{2}}}{a b}$

Step 4.7

Pull terms out from under the radical.

$\frac{\sqrt[18]{a^{9} (b^{4} \sqrt{b}) {(a^{8})}^{2} {(b^{8})}^{2}}}{a b}$

Step 4.8

Multiply the exponents in ${(a^{8})}^{2}$ .

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Step 4.8.1

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

$\frac{\sqrt[18]{a^{9} (b^{4} \sqrt{b}) a^{8 \cdot 2} {(b^{8})}^{2}}}{a b}$

Step 4.8.2

Multiply $8$ by $2$ .

$\frac{\sqrt[18]{a^{9} (b^{4} \sqrt{b}) a^{16} {(b^{8})}^{2}}}{a b}$

Step 4.9

Multiply the exponents in ${(b^{8})}^{2}$ .

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Step 4.9.1

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

$\frac{\sqrt[18]{a^{9} (b^{4} \sqrt{b}) a^{16} b^{8 \cdot 2}}}{a b}$

Step 4.9.2

Multiply $8$ by $2$ .

$\frac{\sqrt[18]{a^{9} (b^{4} \sqrt{b}) a^{16} b^{16}}}{a b}$

$\frac{\sqrt[18]{a^{9} b^{4} \sqrt{b} a^{16} b^{16}}}{a b}$

Step 4.10

Combine exponents.

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Step 4.10.1

Multiply $a^{9}$ by $a^{16}$ by adding the exponents.

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Step 4.10.1.1

Move $a^{16}$ .

$\frac{\sqrt[18]{a^{16} a^{9} b^{4} \sqrt{b} b^{16}}}{a b}$

Step 4.10.1.2

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

$\frac{\sqrt[18]{a^{16 + 9} b^{4} \sqrt{b} b^{16}}}{a b}$

Step 4.10.1.3

Add $16$ and $9$ .

$\frac{\sqrt[18]{a^{25} b^{4} \sqrt{b} b^{16}}}{a b}$

Step 4.10.2

Multiply $b^{4}$ by $b^{16}$ by adding the exponents.

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Step 4.10.2.1

Move $b^{16}$ .

$\frac{\sqrt[18]{a^{25} (b^{16} b^{4}) \sqrt{b}}}{a b}$

Step 4.10.2.2

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

$\frac{\sqrt[18]{a^{25} b^{16 + 4} \sqrt{b}}}{a b}$

Step 4.10.2.3

Add $16$ and $4$ .

$\frac{\sqrt[18]{a^{25} b^{20} \sqrt{b}}}{a b}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $a^{25} b^{20} \sqrt{b}$ as ${(a b)}^{18} (a^{7} b^{2} \sqrt{b})$ .

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Step 5.1.1

Factor out $a^{18}$ .

$\frac{\sqrt[18]{a^{18} a^{7} b^{20} \sqrt{b}}}{a b}$

Step 5.1.2

Factor out $b^{18}$ .

$\frac{\sqrt[18]{a^{18} a^{7} (b^{18} b^{2}) \sqrt{b}}}{a b}$

Step 5.1.3

Move $a^{7}$ .

$\frac{\sqrt[18]{a^{18} b^{18} a^{7} b^{2} \sqrt{b}}}{a b}$

Step 5.1.4

Rewrite $a^{18} b^{18}$ as ${(a b)}^{18}$ .

$\frac{\sqrt[18]{{(a b)}^{18} a^{7} b^{2} \sqrt{b}}}{a b}$

Step 5.1.5

Add parentheses.

$\frac{\sqrt[18]{{(a b)}^{18} a^{7} (b^{2} \sqrt{b})}}{a b}$

Step 5.1.6

Add parentheses.

$\frac{\sqrt[18]{{(a b)}^{18} (a^{7} b^{2} \sqrt{b})}}{a b}$

Step 5.2

Pull terms out from under the radical.

$\frac{a b \sqrt[18]{a^{7} b^{2} \sqrt{b}}}{a b}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $a$ .

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Step 6.1.1

Cancel the common factor.

$\frac{a b \sqrt[18]{a^{7} b^{2} \sqrt{b}}}{a b}$

Step 6.1.2

Rewrite the expression.

$\frac{b \sqrt[18]{a^{7} b^{2} \sqrt{b}}}{b}$

Step 6.2

Cancel the common factor of $b$ .

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Step 6.2.1

Cancel the common factor.

$\frac{b \sqrt[18]{a^{7} b^{2} \sqrt{b}}}{b}$

Step 6.2.2

Divide $\sqrt[18]{a^{7} b^{2} \sqrt{b}}$ by $1$ .

$\sqrt[18]{a^{7} b^{2} \sqrt{b}}$