Simplify the Radical Expression ( square root of y)/( square root of y- square root of 7)

$\frac{\sqrt{y}}{\sqrt{y} - \sqrt{7}}$

Step 1

Multiply $\frac{\sqrt{y}}{\sqrt{y} - \sqrt{7}}$ by $\frac{\sqrt{y} + \sqrt{7}}{\sqrt{y} + \sqrt{7}}$ .

$\frac{\sqrt{y}}{\sqrt{y} - \sqrt{7}} \cdot \frac{\sqrt{y} + \sqrt{7}}{\sqrt{y} + \sqrt{7}}$

Step 2

Multiply $\frac{\sqrt{y}}{\sqrt{y} - \sqrt{7}}$ by $\frac{\sqrt{y} + \sqrt{7}}{\sqrt{y} + \sqrt{7}}$ .

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{(\sqrt{y} - \sqrt{7}) (\sqrt{y} + \sqrt{7})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{{\sqrt{y}}^{2} + \sqrt{y \cdot 7} - \sqrt{y \cdot 7} - {\sqrt{7}}^{2}}$

Step 4

Simplify.

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

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{{(y^{\frac{1}{2}})}^{2} - {\sqrt{7}}^{2}}$

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y^{\frac{1}{2} \cdot 2} - {\sqrt{7}}^{2}}$

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y^{\frac{2}{2}} - {\sqrt{7}}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y^{\frac{2}{2}} - {\sqrt{7}}^{2}}$

Rewrite the expression.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y^{1} - {\sqrt{7}}^{2}}$

Simplify.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y - {\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y - {(7^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y - 7^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y - 7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y - 7^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y - 7^{1}}$

Evaluate the exponent.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y - 1 \cdot 7}$

Multiply $- 1$ by $7$ .

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{7})}{y - 7}$

Step 5

Apply the distributive property.

$\frac{\sqrt{y} \sqrt{y} + \sqrt{y} \sqrt{7}}{y - 7}$

Step 6

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

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

$\frac{{\sqrt{y}}^{1} \sqrt{y} + \sqrt{y} \sqrt{7}}{y - 7}$

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

$\frac{{\sqrt{y}}^{1} {\sqrt{y}}^{1} + \sqrt{y} \sqrt{7}}{y - 7}$

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

$\frac{{\sqrt{y}}^{1 + 1} + \sqrt{y} \sqrt{7}}{y - 7}$

Add $1$ and $1$ .

$\frac{{\sqrt{y}}^{2} + \sqrt{y} \sqrt{7}}{y - 7}$

Step 7

Combine using the product rule for radicals.

$\frac{{\sqrt{y}}^{2} + \sqrt{y \cdot 7}}{y - 7}$

Step 8

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

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

$\frac{{(y^{\frac{1}{2}})}^{2} + \sqrt{y \cdot 7}}{y - 7}$

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

$\frac{y^{\frac{1}{2} \cdot 2} + \sqrt{y \cdot 7}}{y - 7}$

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

$\frac{y^{\frac{2}{2}} + \sqrt{y \cdot 7}}{y - 7}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{y^{\frac{2}{2}} + \sqrt{y \cdot 7}}{y - 7}$

Rewrite the expression.

$\frac{y^{1} + \sqrt{y \cdot 7}}{y - 7}$

Simplify.

$\frac{y + \sqrt{y \cdot 7}}{y - 7}$

Step 9

Reorder factors in $y + \sqrt{y \cdot 7}$ .

$\frac{y + \sqrt{7 y}}{y - 7}$

Step 10

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 y}$ as ${(7 y)}^{\frac{1}{2}}$ .

$\frac{y + {(7 y)}^{\frac{1}{2}}}{y - 7}$

Apply the product rule to $7 y$ .

$\frac{y + 7^{\frac{1}{2}} y^{\frac{1}{2}}}{y - 7}$

Factor $y^{\frac{1}{2}}$ out of $y + 7^{\frac{1}{2}} y^{\frac{1}{2}}$ .

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

$\frac{y^{1} + 7^{\frac{1}{2}} y^{\frac{1}{2}}}{y - 7}$

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

$\frac{y^{\frac{1}{2}} y^{\frac{1}{2}} + 7^{\frac{1}{2}} y^{\frac{1}{2}}}{y - 7}$

Factor $y^{\frac{1}{2}}$ out of $7^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$\frac{y^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}}{y - 7}$

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}$ .

$\frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 7^{\frac{1}{2}})}{y - 7}$

Simplify the Radical Expression ( square root of y)/( square root of y- square root of 7)

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