Algebra Examples

$\frac{\sqrt{100 x^{2}} + \sqrt{2} \sqrt{32 x^{4}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1

Simplify the numerator.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{100 x^{2}}$ as ${(100 x^{2})}^{\frac{1}{2}}$ .

$\frac{{(100 x^{2})}^{\frac{1}{2}} + \sqrt{2} \sqrt{32 x^{4}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.2

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

$\frac{{(100 x^{2})}^{\frac{1}{2}} + 2^{\frac{1}{2}} \sqrt{32 x^{4}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{32 x^{4}}$ as ${(32 x^{4})}^{\frac{1}{2}}$ .

$\frac{{(100 x^{2})}^{\frac{1}{2}} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.4

Apply the product rule to $100 x^{2}$ .

$\frac{100^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.5

Rewrite $100$ as $10^{2}$ .

$\frac{{(10^{2})}^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.6

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

$\frac{10^{2 (\frac{1}{2})} {(x^{2})}^{\frac{1}{2}} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

Cancel the common factor.

$\frac{10^{2 (\frac{1}{2})} {(x^{2})}^{\frac{1}{2}} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.7.2

Rewrite the expression.

$\frac{10^{1} {(x^{2})}^{\frac{1}{2}} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.8

Evaluate the exponent.

$\frac{10 {(x^{2})}^{\frac{1}{2}} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.9

Multiply the exponents in ${(x^{2})}^{\frac{1}{2}}$ .

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Step 1.9.1

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

$\frac{10 x^{2 (\frac{1}{2})} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.9.2

Cancel the common factor of $2$ .

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Step 1.9.2.1

Cancel the common factor.

$\frac{10 x^{2 (\frac{1}{2})} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.9.2.2

Rewrite the expression.

$\frac{10 x^{1} + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.10

Simplify.

$\frac{10 x + 2^{\frac{1}{2}} {(32 x^{4})}^{\frac{1}{2}}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.11

Apply the product rule to $32 x^{4}$ .

$\frac{10 x + 2^{\frac{1}{2}} (32^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.12

Multiply the exponents in ${(x^{4})}^{\frac{1}{2}}$ .

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Step 1.12.1

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

$\frac{10 x + 2^{\frac{1}{2}} (32^{\frac{1}{2}} x^{4 (\frac{1}{2})})}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.12.2

Cancel the common factor of $2$ .

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Step 1.12.2.1

Factor $2$ out of $4$ .

$\frac{10 x + 2^{\frac{1}{2}} (32^{\frac{1}{2}} x^{2 (2) \frac{1}{2}})}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.12.2.2

Cancel the common factor.

$\frac{10 x + 2^{\frac{1}{2}} (32^{\frac{1}{2}} x^{2 \cdot 2 \frac{1}{2}})}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.12.2.3

Rewrite the expression.

$\frac{10 x + 2^{\frac{1}{2}} (32^{\frac{1}{2}} x^{2})}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13

Multiply $2^{\frac{1}{2}} (32^{\frac{1}{2}} x^{2})$ .

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Step 1.13.1

Rewrite $32$ as $2^{5}$ .

$\frac{10 x + {(2^{5})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.2

Multiply the exponents in ${(2^{5})}^{\frac{1}{2}}$ .

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Step 1.13.2.1

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

$\frac{10 x + 2^{5 (\frac{1}{2})} \cdot 2^{\frac{1}{2}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.2.2

Combine $5$ and $\frac{1}{2}$ .

$\frac{10 x + 2^{\frac{5}{2}} \cdot 2^{\frac{1}{2}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.3

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

$\frac{10 x + 2^{\frac{5}{2} + \frac{1}{2}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.4

Combine the numerators over the common denominator.

$\frac{10 x + 2^{\frac{5 + 1}{2}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.5

Add $5$ and $1$ .

$\frac{10 x + 2^{\frac{6}{2}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.6

Cancel the common factor of $6$ and $2$ .

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Step 1.13.6.1

Factor $2$ out of $6$ .

$\frac{10 x + 2^{\frac{2 \cdot 3}{2}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.6.2

Cancel the common factors.

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Step 1.13.6.2.1

Factor $2$ out of $2$ .

$\frac{10 x + 2^{\frac{2 \cdot 3}{2 (1)}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.6.2.2

Cancel the common factor.

$\frac{10 x + 2^{\frac{2 \cdot 3}{2 \cdot 1}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.6.2.3

Rewrite the expression.

$\frac{10 x + 2^{\frac{3}{1}} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.13.6.2.4

Divide $3$ by $1$ .

$\frac{10 x + 2^{3} x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.14

Raise $2$ to the power of $3$ .

$\frac{10 x + 8 x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.15

Factor $2 x$ out of $10 x + 8 x^{2}$ .

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Step 1.15.1

Factor $2 x$ out of $10 x$ .

$\frac{2 x (5) + 8 x^{2}}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.15.2

Factor $2 x$ out of $8 x^{2}$ .

$\frac{2 x (5) + 2 x (4 x)}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 1.15.3

Factor $2 x$ out of $2 x (5) + 2 x (4 x)$ .

$\frac{2 x (5 + 4 x)}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$

Step 2

Multiply $\frac{2 x (5 + 4 x)}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$ by $\frac{\sqrt{x^{2} + 14} + 3 \sqrt{x}}{\sqrt{x^{2} + 14} + 3 \sqrt{x}}$ .

$\frac{2 x (5 + 4 x)}{\sqrt{x^{2} + 14} - 3 \sqrt{x}} \cdot \frac{\sqrt{x^{2} + 14} + 3 \sqrt{x}}{\sqrt{x^{2} + 14} + 3 \sqrt{x}}$

Step 3

Combine fractions.

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Step 3.1

Multiply $\frac{2 x (5 + 4 x)}{\sqrt{x^{2} + 14} - 3 \sqrt{x}}$ by $\frac{\sqrt{x^{2} + 14} + 3 \sqrt{x}}{\sqrt{x^{2} + 14} + 3 \sqrt{x}}$ .

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{(\sqrt{x^{2} + 14} - 3 \sqrt{x}) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{{\sqrt{x^{2} + 14}}^{2} + 3 \sqrt{(x^{2} + 14) x} - 3 \sqrt{(x^{2} + 14) x} - 9 {\sqrt{x}}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Rewrite ${\sqrt{x^{2} + 14}}^{2}$ as $x^{2} + 14$ .

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Step 3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 14}$ as ${(x^{2} + 14)}^{\frac{1}{2}}$ .

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{{({(x^{2} + 14)}^{\frac{1}{2}})}^{2} - 9 {\sqrt{x}}^{2}}$

Step 3.3.1.2

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

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{{(x^{2} + 14)}^{\frac{1}{2} \cdot 2} - 9 {\sqrt{x}}^{2}}$

Step 3.3.1.3

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

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{{(x^{2} + 14)}^{\frac{2}{2}} - 9 {\sqrt{x}}^{2}}$

Step 3.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.4.1

Cancel the common factor.

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{{(x^{2} + 14)}^{\frac{2}{2}} - 9 {\sqrt{x}}^{2}}$

Step 3.3.1.4.2

Rewrite the expression.

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{{(x^{2} + 14)}^{1} - 9 {\sqrt{x}}^{2}}$

Step 3.3.1.5

Simplify.

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{x^{2} + 14 - 9 {\sqrt{x}}^{2}}$

Step 3.3.2

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

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Step 3.3.2.1

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

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{x^{2} + 14 - 9 {(x^{\frac{1}{2}})}^{2}}$

Step 3.3.2.2

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

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{x^{2} + 14 - 9 x^{\frac{1}{2} \cdot 2}}$

Step 3.3.2.3

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

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{x^{2} + 14 - 9 x^{\frac{2}{2}}}$

Step 3.3.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.4.1

Cancel the common factor.

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{x^{2} + 14 - 9 x^{\frac{2}{2}}}$

Step 3.3.2.4.2

Rewrite the expression.

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{x^{2} + 14 - 9 x^{1}}$

Step 3.3.2.5

Simplify.

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{x^{2} + 14 - 9 x}$

Step 4

Factor $x^{2} + 14 - 9 x$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 4.2

Write the factored form using these integers.

$\frac{2 x (5 + 4 x) (\sqrt{x^{2} + 14} + 3 \sqrt{x})}{(x - 7) (x - 2)}$