Simplify 5/( square root of 90)

$\frac{5}{\sqrt{90}}$

Step 1

Simplify the denominator.

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

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

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

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

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{5}{3 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

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

Step 3

Combine and simplify the denominator.

Multiply $\frac{5}{3 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

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

Move $\sqrt{10}$ .

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

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{5 \sqrt{10}}{3 ({\sqrt{10}}^{1} \sqrt{10})}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{5 \sqrt{10}}{3 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})}$

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

$\frac{5 \sqrt{10}}{3 {\sqrt{10}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

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

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

$\frac{5 \sqrt{10}}{3 \cdot 10^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{5 \sqrt{10}}{3 \cdot 10^{1}}$

Evaluate the exponent.

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

Step 4

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

Factor $5$ out of $5 \sqrt{10}$ .

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

Cancel the common factors.

Factor $5$ out of $3 \cdot 10$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Multiply $3$ by $2$ .

$\frac{\sqrt{10}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{10}}{6}$

Decimal Form:

$0.52704627 \dots$

Simplify 5/( square root of 90)

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