Algebra Examples

$\frac{2 + \sqrt{x}}{3 - 2 \sqrt{x}}$

Step 1

Multiply $\frac{2 + \sqrt{x}}{3 - 2 \sqrt{x}}$ by $\frac{3 + 2 \sqrt{x}}{3 + 2 \sqrt{x}}$ .

$\frac{2 + \sqrt{x}}{3 - 2 \sqrt{x}} \cdot \frac{3 + 2 \sqrt{x}}{3 + 2 \sqrt{x}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{2 + \sqrt{x}}{3 - 2 \sqrt{x}}$ by $\frac{3 + 2 \sqrt{x}}{3 + 2 \sqrt{x}}$ .

$\frac{(2 + \sqrt{x}) (3 + 2 \sqrt{x})}{(3 - 2 \sqrt{x}) (3 + 2 \sqrt{x})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(2 + \sqrt{x}) (3 + 2 \sqrt{x})}{9 + 6 \sqrt{x} - 6 \sqrt{x} - 4 {\sqrt{x}}^{2}}$

Step 2.3

Simplify.

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Step 2.3.1

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

$\frac{(2 + \sqrt{x}) (3 + 2 \sqrt{x})}{9 - 4 {(x^{\frac{1}{2}})}^{2}}$

Step 2.3.2

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

$\frac{(2 + \sqrt{x}) (3 + 2 \sqrt{x})}{9 - 4 x^{\frac{1}{2} \cdot 2}}$

Step 2.3.3

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

$\frac{(2 + \sqrt{x}) (3 + 2 \sqrt{x})}{9 - 4 x^{\frac{2}{2}}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\frac{(2 + \sqrt{x}) (3 + 2 \sqrt{x})}{9 - 4 x^{\frac{2}{2}}}$

Step 2.3.4.2

Rewrite the expression.

$\frac{(2 + \sqrt{x}) (3 + 2 \sqrt{x})}{9 - 4 x^{1}}$

Step 2.3.5

Simplify.

$\frac{(2 + \sqrt{x}) (3 + 2 \sqrt{x})}{9 - 4 x}$

Step 3

Expand $(2 + \sqrt{x}) (3 + 2 \sqrt{x})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{2 (3 + 2 \sqrt{x}) + \sqrt{x} (3 + 2 \sqrt{x})}{9 - 4 x}$

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (2 \sqrt{x}) + \sqrt{x} (3 + 2 \sqrt{x})}{9 - 4 x}$

Step 3.3

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (2 \sqrt{x}) + \sqrt{x} \cdot 3 + \sqrt{x} (2 \sqrt{x})}{9 - 4 x}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $2$ by $3$ .

$\frac{6 + 2 (2 \sqrt{x}) + \sqrt{x} \cdot 3 + \sqrt{x} (2 \sqrt{x})}{9 - 4 x}$

Step 4.1.2

Multiply $2$ by $2$ .

$\frac{6 + 4 \sqrt{x} + \sqrt{x} \cdot 3 + \sqrt{x} (2 \sqrt{x})}{9 - 4 x}$

Step 4.1.3

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

$\frac{6 + 4 \sqrt{x} + 3 \cdot \sqrt{x} + \sqrt{x} (2 \sqrt{x})}{9 - 4 x}$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 \sqrt{x} \sqrt{x}}{9 - 4 x}$

Step 4.1.5

Multiply $2 \sqrt{x} \sqrt{x}$ .

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Step 4.1.5.1

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

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 ({\sqrt{x}}^{1} \sqrt{x})}{9 - 4 x}$

Step 4.1.5.2

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

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 ({\sqrt{x}}^{1} {\sqrt{x}}^{1})}{9 - 4 x}$

Step 4.1.5.3

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

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 {\sqrt{x}}^{1 + 1}}{9 - 4 x}$

Step 4.1.5.4

Add $1$ and $1$ .

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 {\sqrt{x}}^{2}}{9 - 4 x}$

Step 4.1.6

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

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Step 4.1.6.1

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

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 {(x^{\frac{1}{2}})}^{2}}{9 - 4 x}$

Step 4.1.6.2

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

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 x^{\frac{1}{2} \cdot 2}}{9 - 4 x}$

Step 4.1.6.3

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

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 x^{\frac{2}{2}}}{9 - 4 x}$

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 x^{\frac{2}{2}}}{9 - 4 x}$

Step 4.1.6.4.2

Rewrite the expression.

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 x^{1}}{9 - 4 x}$

Step 4.1.6.5

Simplify.

$\frac{6 + 4 \sqrt{x} + 3 \sqrt{x} + 2 x}{9 - 4 x}$

Step 4.2

Add $4 \sqrt{x}$ and $3 \sqrt{x}$ .

$\frac{6 + 7 \sqrt{x} + 2 x}{9 - 4 x}$