Simplify ( square root of y)/( square root of y- square root of 6)

$\frac{\sqrt{y}}{\sqrt{y} - \sqrt{6}}$

Step 1

Multiply $\frac{\sqrt{y}}{\sqrt{y} - \sqrt{6}}$ by $\frac{\sqrt{y} + \sqrt{6}}{\sqrt{y} + \sqrt{6}}$ .

$\frac{\sqrt{y}}{\sqrt{y} - \sqrt{6}} \cdot \frac{\sqrt{y} + \sqrt{6}}{\sqrt{y} + \sqrt{6}}$

Step 2

Multiply $\frac{\sqrt{y}}{\sqrt{y} - \sqrt{6}}$ by $\frac{\sqrt{y} + \sqrt{6}}{\sqrt{y} + \sqrt{6}}$ .

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{(\sqrt{y} - \sqrt{6}) (\sqrt{y} + \sqrt{6})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{{\sqrt{y}}^{2} + \sqrt{y \cdot 6} - \sqrt{y \cdot 6} - {\sqrt{6}}^{2}}$

Step 4

Simplify.

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

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{{(y^{\frac{1}{2}})}^{2} - {\sqrt{6}}^{2}}$

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y^{\frac{1}{2} \cdot 2} - {\sqrt{6}}^{2}}$

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y^{\frac{2}{2}} - {\sqrt{6}}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y^{\frac{2}{2}} - {\sqrt{6}}^{2}}$

Rewrite the expression.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y^{1} - {\sqrt{6}}^{2}}$

Simplify.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y - {\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y - {(6^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y - 6^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y - 6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y - 6^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y - 6^{1}}$

Evaluate the exponent.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y - 1 \cdot 6}$

Multiply $- 1$ by $6$ .

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{6})}{y - 6}$

Step 5

Apply the distributive property.

$\frac{\sqrt{y} \sqrt{y} + \sqrt{y} \sqrt{6}}{y - 6}$

Step 6

Multiply $\sqrt{y} \sqrt{y}$ .

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

$\frac{{\sqrt{y}}^{1} \sqrt{y} + \sqrt{y} \sqrt{6}}{y - 6}$

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

$\frac{{\sqrt{y}}^{1} {\sqrt{y}}^{1} + \sqrt{y} \sqrt{6}}{y - 6}$

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

$\frac{{\sqrt{y}}^{1 + 1} + \sqrt{y} \sqrt{6}}{y - 6}$

Add $1$ and $1$ .

$\frac{{\sqrt{y}}^{2} + \sqrt{y} \sqrt{6}}{y - 6}$

Step 7

Simplify terms.

Combine using the product rule for radicals.

$\frac{{\sqrt{y}}^{2} + \sqrt{y \cdot 6}}{y - 6}$

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

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

$\frac{{(y^{\frac{1}{2}})}^{2} + \sqrt{y \cdot 6}}{y - 6}$

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

$\frac{y^{\frac{1}{2} \cdot 2} + \sqrt{y \cdot 6}}{y - 6}$

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

$\frac{y^{\frac{2}{2}} + \sqrt{y \cdot 6}}{y - 6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{y^{\frac{2}{2}} + \sqrt{y \cdot 6}}{y - 6}$

Rewrite the expression.

$\frac{y^{1} + \sqrt{y \cdot 6}}{y - 6}$

Simplify.

$\frac{y + \sqrt{y \cdot 6}}{y - 6}$

Reorder factors in $y + \sqrt{y \cdot 6}$ .

$\frac{y + \sqrt{6 y}}{y - 6}$

Step 8

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 y}$ as ${(6 y)}^{\frac{1}{2}}$ .

$\frac{y + {(6 y)}^{\frac{1}{2}}}{y - 6}$

Apply the product rule to $6 y$ .

$\frac{y + 6^{\frac{1}{2}} y^{\frac{1}{2}}}{y - 6}$

Factor $y^{\frac{1}{2}}$ out of $y + 6^{\frac{1}{2}} y^{\frac{1}{2}}$ .

Raise $y$ to the power of $1$ .

$\frac{y^{1} + 6^{\frac{1}{2}} y^{\frac{1}{2}}}{y - 6}$

Factor $y^{\frac{1}{2}}$ out of $y^{1}$ .

$\frac{y^{\frac{1}{2}} y^{\frac{1}{2}} + 6^{\frac{1}{2}} y^{\frac{1}{2}}}{y - 6}$

Factor $y^{\frac{1}{2}}$ out of $6^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$\frac{y^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} \cdot 6^{\frac{1}{2}}}{y - 6}$

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} \cdot 6^{\frac{1}{2}}$ .

$\frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 6^{\frac{1}{2}})}{y - 6}$

Simplify ( square root of y)/( square root of y- square root of 6)

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