Evaluate ( square root of 10)/( square root of 15)

$\frac{\sqrt{10}}{\sqrt{15}}$

Step 1

Combine $\sqrt{10}$ and $\sqrt{15}$ into a single radical.

$\sqrt{\frac{10}{15}}$

Step 2

Cancel the common factor of $10$ and $15$ .

Factor $5$ out of $10$ .

$\sqrt{\frac{5 (2)}{15}}$

Cancel the common factors.

Factor $5$ out of $15$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 6

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{3}$

Decimal Form:

$0.81649658 \dots$

Evaluate ( square root of 10)/( square root of 15)

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