Simplify ( square root of b)/(3+ square root of b)

$\frac{\sqrt{b}}{3 + \sqrt{b}}$

Step 1

Multiply $\frac{\sqrt{b}}{3 + \sqrt{b}}$ by $\frac{3 - \sqrt{b}}{3 - \sqrt{b}}$ .

$\frac{\sqrt{b}}{3 + \sqrt{b}} \cdot \frac{3 - \sqrt{b}}{3 - \sqrt{b}}$

Step 2

Simplify terms.

Multiply $\frac{\sqrt{b}}{3 + \sqrt{b}}$ by $\frac{3 - \sqrt{b}}{3 - \sqrt{b}}$ .

$\frac{\sqrt{b} (3 - \sqrt{b})}{(3 + \sqrt{b}) (3 - \sqrt{b})}$

Simplify terms.

Expand the denominator using the FOIL method.

$\frac{\sqrt{b} (3 - \sqrt{b})}{9 - 3 \sqrt{b} + \sqrt{b} \cdot 3 - {\sqrt{b}}^{2}}$

Simplify.

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

$\frac{\sqrt{b} (3 - \sqrt{b})}{9 - {(b^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{b} (3 - \sqrt{b})}{9 - b^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{b} (3 - \sqrt{b})}{9 - b^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{b} (3 - \sqrt{b})}{9 - b^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{b} (3 - \sqrt{b})}{9 - b^{1}}$

Simplify.

$\frac{\sqrt{b} (3 - \sqrt{b})}{9 - b}$

Apply the distributive property.

$\frac{\sqrt{b} \cdot 3 + \sqrt{b} (- \sqrt{b})}{9 - b}$

Reorder.

Move $3$ to the left of $\sqrt{b}$ .

$\frac{3 \cdot \sqrt{b} + \sqrt{b} (- \sqrt{b})}{9 - b}$

Rewrite using the commutative property of multiplication.

$\frac{3 \cdot \sqrt{b} - \sqrt{b} \sqrt{b}}{9 - b}$

Simplify each term.

Multiply $- \sqrt{b} \sqrt{b}$ .

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

$\frac{3 \sqrt{b} - ({\sqrt{b}}^{1} \sqrt{b})}{9 - b}$

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

$\frac{3 \sqrt{b} - ({\sqrt{b}}^{1} {\sqrt{b}}^{1})}{9 - b}$

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

$\frac{3 \sqrt{b} - {\sqrt{b}}^{1 + 1}}{9 - b}$

Add $1$ and $1$ .

$\frac{3 \sqrt{b} - {\sqrt{b}}^{2}}{9 - b}$

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

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

$\frac{3 \sqrt{b} - {(b^{\frac{1}{2}})}^{2}}{9 - b}$

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

$\frac{3 \sqrt{b} - b^{\frac{1}{2} \cdot 2}}{9 - b}$

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

$\frac{3 \sqrt{b} - b^{\frac{2}{2}}}{9 - b}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 \sqrt{b} - b^{\frac{2}{2}}}{9 - b}$

Rewrite the expression.

$\frac{3 \sqrt{b} - b^{1}}{9 - b}$

Simplify.

$\frac{3 \sqrt{b} - b}{9 - b}$

Step 3

Simplify the numerator.

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

$\frac{3 b^{\frac{1}{2}} - b}{9 - b}$

Factor $b^{\frac{1}{2}}$ out of $3 b^{\frac{1}{2}} - b$ .

Factor $b^{\frac{1}{2}}$ out of $3 b^{\frac{1}{2}}$ .

$\frac{b^{\frac{1}{2}} \cdot 3 - b}{9 - b}$

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

$\frac{b^{\frac{1}{2}} \cdot 3 + b^{\frac{1}{2}} (- b^{\frac{1}{2}})}{9 - b}$

Factor $b^{\frac{1}{2}}$ out of $b^{\frac{1}{2}} \cdot 3 + b^{\frac{1}{2}} (- b^{\frac{1}{2}})$ .

$\frac{b^{\frac{1}{2}} (3 - b^{\frac{1}{2}})}{9 - b}$

Simplify ( square root of b)/(3+ square root of b)

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