Simplify ( square root of a)/( square root of a+ square root of b)

$\frac{\sqrt{a}}{\sqrt{a} + \sqrt{b}}$

Step 1

Multiply $\frac{\sqrt{a}}{\sqrt{a} + \sqrt{b}}$ by $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$ .

$\frac{\sqrt{a}}{\sqrt{a} + \sqrt{b}} \cdot \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$

Step 2

Multiply $\frac{\sqrt{a}}{\sqrt{a} + \sqrt{b}}$ by $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$ .

$\frac{\sqrt{a} (\sqrt{a} - \sqrt{b})}{(\sqrt{a} + \sqrt{b}) (\sqrt{a} - \sqrt{b})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{a} (\sqrt{a} - \sqrt{b})}{{\sqrt{a}}^{2} - \sqrt{a b} + \sqrt{b a} - {\sqrt{b}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{a} (\sqrt{a} - \sqrt{b})}{a - b}$

Step 5

Apply the distributive property.

$\frac{\sqrt{a} \sqrt{a} + \sqrt{a} (- \sqrt{b})}{a - b}$

Step 6

Multiply $\sqrt{a} \sqrt{a}$ .

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

$\frac{{\sqrt{a}}^{1} \sqrt{a} + \sqrt{a} (- \sqrt{b})}{a - b}$

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

$\frac{{\sqrt{a}}^{1} {\sqrt{a}}^{1} + \sqrt{a} (- \sqrt{b})}{a - b}$

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

$\frac{{\sqrt{a}}^{1 + 1} + \sqrt{a} (- \sqrt{b})}{a - b}$

Add $1$ and $1$ .

$\frac{{\sqrt{a}}^{2} + \sqrt{a} (- \sqrt{b})}{a - b}$

Step 7

Simplify terms.

Rewrite using the commutative property of multiplication.

$\frac{{\sqrt{a}}^{2} - \sqrt{a} \sqrt{b}}{a - b}$

Simplify each term.

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

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

$\frac{{(a^{\frac{1}{2}})}^{2} - \sqrt{a} \sqrt{b}}{a - b}$

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

$\frac{a^{\frac{1}{2} \cdot 2} - \sqrt{a} \sqrt{b}}{a - b}$

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

$\frac{a^{\frac{2}{2}} - \sqrt{a} \sqrt{b}}{a - b}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{a^{\frac{2}{2}} - \sqrt{a} \sqrt{b}}{a - b}$

Rewrite the expression.

$\frac{a^{1} - \sqrt{a} \sqrt{b}}{a - b}$

Simplify.

$\frac{a - \sqrt{a} \sqrt{b}}{a - b}$

Combine using the product rule for radicals.

$\frac{a - \sqrt{b a}}{a - b}$

Step 8

Simplify the numerator.

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

$\frac{a - {(b a)}^{\frac{1}{2}}}{a - b}$

Apply the product rule to $b a$ .

$\frac{a - (b^{\frac{1}{2}} a^{\frac{1}{2}})}{a - b}$

Factor $a^{\frac{1}{2}}$ out of $a - b^{\frac{1}{2}} a^{\frac{1}{2}}$ .

Move $b^{\frac{1}{2}}$ .

$\frac{a - a^{\frac{1}{2}} b^{\frac{1}{2}}}{a - b}$

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

$\frac{a^{1} - a^{\frac{1}{2}} b^{\frac{1}{2}}}{a - b}$

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

$\frac{a^{\frac{1}{2}} a^{\frac{1}{2}} - a^{\frac{1}{2}} b^{\frac{1}{2}}}{a - b}$

Factor $a^{\frac{1}{2}}$ out of $- a^{\frac{1}{2}} b^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} a^{\frac{1}{2}} + a^{\frac{1}{2}} (- b^{\frac{1}{2}})}{a - b}$

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} a^{\frac{1}{2}} + a^{\frac{1}{2}} (- b^{\frac{1}{2}})$ .

$\frac{a^{\frac{1}{2}} (a^{\frac{1}{2}} - b^{\frac{1}{2}})}{a - b}$

Simplify ( square root of a)/( square root of a+ square root of b)

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