Algebra Examples

$\frac{\sqrt{125} + \sqrt{20} - \sqrt{45}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 1.1.1

Factor $25$ out of $125$ .

$\frac{\sqrt{25 (5)} + \sqrt{20} - \sqrt{45}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2} \cdot 5} + \sqrt{20} - \sqrt{45}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.2

Pull terms out from under the radical.

$\frac{5 \sqrt{5} + \sqrt{20} - \sqrt{45}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.3

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

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Step 1.3.1

Factor $4$ out of $20$ .

$\frac{5 \sqrt{5} + \sqrt{4 (5)} - \sqrt{45}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.3.2

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

$\frac{5 \sqrt{5} + \sqrt{2^{2} \cdot 5} - \sqrt{45}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.4

Pull terms out from under the radical.

$\frac{5 \sqrt{5} + 2 \sqrt{5} - \sqrt{45}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.5

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 1.5.1

Factor $9$ out of $45$ .

$\frac{5 \sqrt{5} + 2 \sqrt{5} - \sqrt{9 (5)}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.5.2

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

$\frac{5 \sqrt{5} + 2 \sqrt{5} - \sqrt{3^{2} \cdot 5}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.6

Pull terms out from under the radical.

$\frac{5 \sqrt{5} + 2 \sqrt{5} - (3 \sqrt{5})}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.7

Multiply $3$ by $- 1$ .

$\frac{5 \sqrt{5} + 2 \sqrt{5} - 3 \sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.8

Add $5 \sqrt{5}$ and $2 \sqrt{5}$ .

$\frac{7 \sqrt{5} - 3 \sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 1.9

Subtract $3 \sqrt{5}$ from $7 \sqrt{5}$ .

$\frac{4 \sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{2}}{3}$

Step 2

Combine $\sqrt{5}$ and $\sqrt{10}$ into a single radical.

$4 \sqrt{\frac{5}{10}} \cdot \frac{\sqrt{2}}{3}$

Step 3

Cancel the common factor of $5$ and $10$ .

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Step 3.1

Factor $5$ out of $5$ .

$4 \sqrt{\frac{5 (1)}{10}} \cdot \frac{\sqrt{2}}{3}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $5$ out of $10$ .

$4 \sqrt{\frac{5 \cdot 1}{5 \cdot 2}} \cdot \frac{\sqrt{2}}{3}$

Step 3.2.2

Cancel the common factor.

$4 \sqrt{\frac{5 \cdot 1}{5 \cdot 2}} \cdot \frac{\sqrt{2}}{3}$

Step 3.2.3

Rewrite the expression.

$4 \sqrt{\frac{1}{2}} \cdot \frac{\sqrt{2}}{3}$

Step 4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$4 \frac{\sqrt{1}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{3}$

Step 5

Any root of $1$ is $1$ .

$4 \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{3}$

Step 6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$4 (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) \cdot \frac{\sqrt{2}}{3}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$4 \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \frac{\sqrt{2}}{3}$

Step 7.2

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

$4 \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \cdot \frac{\sqrt{2}}{3}$

Step 7.3

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

$4 \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \cdot \frac{\sqrt{2}}{3}$

Step 7.4

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

$4 \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} \cdot \frac{\sqrt{2}}{3}$

Step 7.5

Add $1$ and $1$ .

$4 \frac{\sqrt{2}}{{\sqrt{2}}^{2}} \cdot \frac{\sqrt{2}}{3}$

Step 7.6

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

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Step 7.6.1

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

$4 \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \cdot \frac{\sqrt{2}}{3}$

Step 7.6.2

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

$4 \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \cdot \frac{\sqrt{2}}{3}$

Step 7.6.3

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

$4 \frac{\sqrt{2}}{2^{\frac{2}{2}}} \cdot \frac{\sqrt{2}}{3}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$4 \frac{\sqrt{2}}{2^{\frac{2}{2}}} \cdot \frac{\sqrt{2}}{3}$

Step 7.6.4.2

Rewrite the expression.

$4 \frac{\sqrt{2}}{2^{1}} \cdot \frac{\sqrt{2}}{3}$

Step 7.6.5

Evaluate the exponent.

$4 \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{3}$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $4$ .

$2 (2) \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{3}$

Step 8.2

Cancel the common factor.

$2 \cdot 2 \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{3}$

Step 8.3

Rewrite the expression.

$2 \sqrt{2} \cdot \frac{\sqrt{2}}{3}$

Step 9

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

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Step 9.1

Combine $\frac{\sqrt{2}}{3}$ and $2$ .

$\frac{\sqrt{2} \cdot 2}{3} \sqrt{2}$

Step 9.2

Combine $\frac{\sqrt{2} \cdot 2}{3}$ and $\sqrt{2}$ .

$\frac{\sqrt{2} \cdot 2 \sqrt{2}}{3}$

Step 9.3

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

$\frac{2 ({\sqrt{2}}^{1} \sqrt{2})}{3}$

Step 9.4

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

$\frac{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{3}$

Step 9.5

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

$\frac{2 {\sqrt{2}}^{1 + 1}}{3}$

Step 9.6

Add $1$ and $1$ .

$\frac{2 {\sqrt{2}}^{2}}{3}$

Step 10

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

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Step 10.1

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

$\frac{2 {(2^{\frac{1}{2}})}^{2}}{3}$

Step 10.2

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

$\frac{2 \cdot 2^{\frac{1}{2} \cdot 2}}{3}$

Step 10.3

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

$\frac{2 \cdot 2^{\frac{2}{2}}}{3}$

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

$\frac{2 \cdot 2^{\frac{2}{2}}}{3}$

Step 10.4.2

Rewrite the expression.

$\frac{2 \cdot 2^{1}}{3}$

Step 10.5

Evaluate the exponent.

$\frac{2 \cdot 2}{3}$

Step 11

Multiply $2$ by $2$ .

$\frac{4}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$

Mixed Number Form:

$1 \frac{1}{3}$