Algebra Examples

$\frac{10 \sqrt{18}}{\sqrt{5}}$

Step 1

Simplify the numerator.

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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 $18$ .

$\frac{10 \sqrt{9 (2)}}{\sqrt{5}}$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

$\frac{10 (| 3 | \sqrt{2})}{\sqrt{5}}$

Step 1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Step 1.4

Multiply $10$ by $3$ .

$\frac{30 \sqrt{2}}{\sqrt{5}}$

Step 2

Multiply $\frac{30 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{30 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{30 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{30 \sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{30 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{30 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 3.4

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

$\frac{30 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{30 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.6.1

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

$\frac{30 \sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{30 \sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{30 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{30 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{30 \sqrt{2} \sqrt{5}}{5^{1}}$

Step 3.6.5

Evaluate the exponent.

$\frac{30 \sqrt{2} \sqrt{5}}{5}$

Step 4

Cancel the common factor of $30$ and $5$ .

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Step 4.1

Factor $5$ out of $30 \sqrt{2} \sqrt{5}$ .

$\frac{5 (6 \sqrt{2} \sqrt{5})}{5}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $5$ out of $5$ .

$\frac{5 (6 \sqrt{2} \sqrt{5})}{5 (1)}$

Step 4.2.2

Cancel the common factor.

$\frac{5 (6 \sqrt{2} \sqrt{5})}{5 \cdot 1}$

Step 4.2.3

Rewrite the expression.

$\frac{6 \sqrt{2} \sqrt{5}}{1}$

Step 4.2.4

Divide $6 \sqrt{2} \sqrt{5}$ by $1$ .

$6 \sqrt{2} \sqrt{5}$

Step 5

Multiply $6 \sqrt{2} \sqrt{5}$ .

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Step 5.1

Combine using the product rule for radicals.

$6 \sqrt{5 \cdot 2}$

Step 5.2

Multiply $5$ by $2$ .

$6 \sqrt{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{10}$

Decimal Form:

$18.97366596 \dots$