Algebra Examples

$\frac{\sqrt{x y}}{\sqrt{3 x}}$

Step 1

Combine $\sqrt{x y}$ and $\sqrt{3 x}$ into a single radical.

$\sqrt{\frac{x y}{3 x}}$

Step 2

Reduce the expression $\frac{x y}{3 x}$ by cancelling the common factors.

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Step 2.1

Cancel the common factor.

$\sqrt{\frac{x y}{3 x}}$

Step 2.2

Rewrite the expression.

$\sqrt{\frac{y}{3}}$

Step 3

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

$\frac{\sqrt{y}}{\sqrt{3}}$

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

$\frac{\sqrt{y} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

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

Step 5.6.4.2

Rewrite the expression.

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

Step 5.6.5

Evaluate the exponent.

$\frac{\sqrt{y} \sqrt{3}}{3}$

Step 6

Combine using the product rule for radicals.

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

Step 7

Reorder factors in $\frac{\sqrt{y \cdot 3}}{3}$ .

$\frac{\sqrt{3 y}}{3}$