Algebra Examples

$\frac{\sqrt{x} + \sqrt{2}}{{(\sqrt{x})}^{3} + 8} \cdot (\frac{x - 2 \sqrt{x} + 4}{\sqrt{x} - 2})$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $8$ as $2^{3}$ .

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

Step 1.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = \sqrt{x}$ and $b = 2$ .

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

Step 1.3

Simplify.

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Step 1.3.1

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

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.1.4

Cancel the common factor of $2$ .

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Step 1.3.1.4.1

Cancel the common factor.

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

Step 1.3.1.4.2

Rewrite the expression.

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

Step 1.3.1.5

Simplify.

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

Step 1.3.2

Multiply $2$ by $- 1$ .

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

Step 1.3.3

Raise $2$ to the power of $2$ .

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $x - 2 \sqrt{x} + 4$ .

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Step 2.1.1

Factor $x - 2 \sqrt{x} + 4$ out of $(\sqrt{x} + 2) (x - 2 \sqrt{x} + 4)$ .

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

Step 2.1.2

Cancel the common factor.

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

Step 2.1.3

Rewrite the expression.

$\frac{\sqrt{x} + \sqrt{2}}{\sqrt{x} + 2} \cdot \frac{1}{\sqrt{x} - 2}$

Step 2.2

Multiply $\frac{\sqrt{x} + \sqrt{2}}{\sqrt{x} + 2}$ by $\frac{1}{\sqrt{x} - 2}$ .

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $\sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 2 + 2 \sqrt{x} + 2 \cdot - 2$ .

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Step 4.1.1

Reorder the factors in the terms $\sqrt{x} \cdot - 2$ and $2 \sqrt{x}$ .

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

Step 4.1.2

Add $- 2 \sqrt{x}$ and $2 \sqrt{x}$ .

$\frac{\sqrt{x} + \sqrt{2}}{\sqrt{x} \sqrt{x} + 0 + 2 \cdot - 2}$

Step 4.1.3

Add $\sqrt{x} \sqrt{x}$ and $0$ .

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

Add $1$ and $1$ .

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

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

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

Step 4.2.2.4

Cancel the common factor of $2$ .

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Step 4.2.2.4.1

Cancel the common factor.

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

Step 4.2.2.4.2

Rewrite the expression.

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

Step 4.2.2.5

Simplify.

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

Step 4.2.3

Multiply $2$ by $- 2$ .

$\frac{\sqrt{x} + \sqrt{2}}{x - 4}$