Algebra Examples

$\frac{9 \sqrt{2}}{\sqrt{10} + \sqrt{18}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 1.1.1

Factor $9$ out of $18$ .

$\frac{9 \sqrt{2}}{\sqrt{10} + \sqrt{9 (2)}}$

Step 1.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{9 \sqrt{2}}{\sqrt{10} + \sqrt{3^{2} \cdot 2}}$

Step 1.2

Pull terms out from under the radical.

$\frac{9 \sqrt{2}}{\sqrt{10} + 3 \sqrt{2}}$

Step 2

Multiply $\frac{9 \sqrt{2}}{\sqrt{10} + 3 \sqrt{2}}$ by $\frac{\sqrt{10} - 3 \sqrt{2}}{\sqrt{10} - 3 \sqrt{2}}$ .

$\frac{9 \sqrt{2}}{\sqrt{10} + 3 \sqrt{2}} \cdot \frac{\sqrt{10} - 3 \sqrt{2}}{\sqrt{10} - 3 \sqrt{2}}$

Step 3

Multiply $\frac{9 \sqrt{2}}{\sqrt{10} + 3 \sqrt{2}}$ by $\frac{\sqrt{10} - 3 \sqrt{2}}{\sqrt{10} - 3 \sqrt{2}}$ .

$\frac{9 \sqrt{2} (\sqrt{10} - 3 \sqrt{2})}{(\sqrt{10} + 3 \sqrt{2}) (\sqrt{10} - 3 \sqrt{2})}$

Step 4

Expand the denominator using the FOIL method.

$\frac{9 \sqrt{2} (\sqrt{10} - 3 \sqrt{2})}{{\sqrt{10}}^{2} - 3 \sqrt{20} + 3 \sqrt{20} - 9 {\sqrt{2}}^{2}}$

Step 5

Simplify.

$\frac{9 \sqrt{2} (\sqrt{10} - 3 \sqrt{2})}{- 8}$

Step 6

Group $\sqrt{10} - 3 \sqrt{2}$ and $\sqrt{2}$ together.

$\frac{9 ((\sqrt{10} - 3 \sqrt{2}) \sqrt{2})}{- 8}$

Step 7

Apply the distributive property.

$\frac{9 (\sqrt{10} \sqrt{2} - 3 \sqrt{2} \sqrt{2})}{- 8}$

Step 8

Combine using the product rule for radicals.

$\frac{9 (\sqrt{10 \cdot 2} - 3 \sqrt{2} \sqrt{2})}{- 8}$

Step 9

Multiply $- 3 \sqrt{2} \sqrt{2}$ .

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Step 9.1

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{9 (\sqrt{10 \cdot 2} - 3 ({\sqrt{2}}^{1} \sqrt{2}))}{- 8}$

Step 9.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{9 (\sqrt{10 \cdot 2} - 3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}))}{- 8}$

Step 9.3

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

$\frac{9 (\sqrt{10 \cdot 2} - 3 {\sqrt{2}}^{1 + 1})}{- 8}$

Step 9.4

Add $1$ and $1$ .

$\frac{9 (\sqrt{10 \cdot 2} - 3 {\sqrt{2}}^{2})}{- 8}$

Step 10

Simplify each term.

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Step 10.1

Multiply $10$ by $2$ .

$\frac{9 (\sqrt{20} - 3 {\sqrt{2}}^{2})}{- 8}$

Step 10.2

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 10.2.1

Factor $4$ out of $20$ .

$\frac{9 (\sqrt{4 (5)} - 3 {\sqrt{2}}^{2})}{- 8}$

Step 10.2.2

Rewrite $4$ as $2^{2}$ .

$\frac{9 (\sqrt{2^{2} \cdot 5} - 3 {\sqrt{2}}^{2})}{- 8}$

Step 10.3

Pull terms out from under the radical.

$\frac{9 (2 \sqrt{5} - 3 {\sqrt{2}}^{2})}{- 8}$

Step 10.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 10.4.1

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

$\frac{9 (2 \sqrt{5} - 3 {(2^{\frac{1}{2}})}^{2})}{- 8}$

Step 10.4.2

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

$\frac{9 (2 \sqrt{5} - 3 \cdot 2^{\frac{1}{2} \cdot 2})}{- 8}$

Step 10.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{9 (2 \sqrt{5} - 3 \cdot 2^{\frac{2}{2}})}{- 8}$

Step 10.4.4

Cancel the common factor of $2$ .

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Step 10.4.4.1

Cancel the common factor.

$\frac{9 (2 \sqrt{5} - 3 \cdot 2^{\frac{2}{2}})}{- 8}$

Step 10.4.4.2

Rewrite the expression.

$\frac{9 (2 \sqrt{5} - 3 \cdot 2^{1})}{- 8}$

Step 10.4.5

Evaluate the exponent.

$\frac{9 (2 \sqrt{5} - 3 \cdot 2)}{- 8}$

Step 10.5

Multiply $- 3$ by $2$ .

$\frac{9 (2 \sqrt{5} - 6)}{- 8}$

Step 11

Cancel the common factor of $2 \sqrt{5} - 6$ and $- 8$ .

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Step 11.1

Factor $2$ out of $9 (2 \sqrt{5} - 6)$ .

$\frac{2 (9 (\sqrt{5} - 3))}{- 8}$

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $2$ out of $- 8$ .

$\frac{2 (9 (\sqrt{5} - 3))}{2 (- 4)}$

Step 11.2.2

Cancel the common factor.

$\frac{2 (9 (\sqrt{5} - 3))}{2 \cdot - 4}$

Step 11.2.3

Rewrite the expression.

$\frac{9 (\sqrt{5} - 3)}{- 4}$

Step 12

Move the negative in front of the fraction.

$- \frac{9 (\sqrt{5} - 3)}{4}$