Algebra Examples

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

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.

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Step 2.1

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

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

Step 2.2

Simplify terms.

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Step 2.2.1

Expand the denominator using the FOIL method.

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

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

Step 2.2.2.4.2

Rewrite the expression.

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

Step 2.2.2.5

Simplify.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.2.4

Reorder.

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Step 2.2.4.1

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

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

Step 2.2.4.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Simplify each term.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.1.4

Add $1$ and $1$ .

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

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.2.4

Cancel the common factor of $2$ .

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Step 2.3.2.4.1

Cancel the common factor.

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

Step 2.3.2.4.2

Rewrite the expression.

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

Step 2.3.2.5

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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