Algebra Examples

$\frac{\sqrt{4 x - 16}}{\sqrt[4]{{(x - 4)}^{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4$ out of $4 x - 16$ .

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Step 1.1.1

Factor $4$ out of $4 x$ .

$\frac{\sqrt{4 (x) - 16}}{\sqrt[4]{{(x - 4)}^{3}}}$

Step 1.1.2

Factor $4$ out of $- 16$ .

$\frac{\sqrt{4 x + 4 \cdot - 4}}{\sqrt[4]{{(x - 4)}^{3}}}$

Step 1.1.3

Factor $4$ out of $4 x + 4 \cdot - 4$ .

$\frac{\sqrt{4 (x - 4)}}{\sqrt[4]{{(x - 4)}^{3}}}$

Step 1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} (x - 4)}}{\sqrt[4]{{(x - 4)}^{3}}}$

Step 1.3

Pull terms out from under the radical.

$\frac{2 \sqrt{x - 4}}{\sqrt[4]{{(x - 4)}^{3}}}$

Step 2

Multiply $\frac{2 \sqrt{x - 4}}{\sqrt[4]{{(x - 4)}^{3}}}$ by $\frac{{\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{\sqrt[4]{{(x - 4)}^{3}}}^{3}}$ .

$\frac{2 \sqrt{x - 4}}{\sqrt[4]{{(x - 4)}^{3}}} \cdot \frac{{\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{\sqrt[4]{{(x - 4)}^{3}}}^{3}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{2 \sqrt{x - 4}}{\sqrt[4]{{(x - 4)}^{3}}}$ by $\frac{{\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{\sqrt[4]{{(x - 4)}^{3}}}^{3}}$ .

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{\sqrt[4]{{(x - 4)}^{3}} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}$

Step 3.2

Raise $\sqrt[4]{{(x - 4)}^{3}}$ to the power of $1$ .

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{\sqrt[4]{{(x - 4)}^{3}}}^{1} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}$

Step 3.3

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

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

Step 3.4

Add $1$ and $3$ .

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{\sqrt[4]{{(x - 4)}^{3}}}^{4}}$

Step 3.5

Rewrite ${\sqrt[4]{{(x - 4)}^{3}}}^{4}$ as ${(x - 4)}^{3}$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{{(x - 4)}^{3}}$ as ${(x - 4)}^{\frac{3}{4}}$ .

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{({(x - 4)}^{\frac{3}{4}})}^{4}}$

Step 3.5.2

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

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{(x - 4)}^{\frac{3}{4} \cdot 4}}$

Step 3.5.3

Combine $\frac{3}{4}$ and $4$ .

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{(x - 4)}^{\frac{3 \cdot 4}{4}}}$

Step 3.5.4

Multiply $3$ by $4$ .

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{(x - 4)}^{\frac{12}{4}}}$

Step 3.5.5

Cancel the common factor of $12$ and $4$ .

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Step 3.5.5.1

Factor $4$ out of $12$ .

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{(x - 4)}^{\frac{4 \cdot 3}{4}}}$

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Factor $4$ out of $4$ .

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

Step 3.5.5.2.2

Cancel the common factor.

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

Step 3.5.5.2.3

Rewrite the expression.

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{(x - 4)}^{\frac{3}{1}}}$

Step 3.5.5.2.4

Divide $3$ by $1$ .

$\frac{2 \sqrt{x - 4} {\sqrt[4]{{(x - 4)}^{3}}}^{3}}{{(x - 4)}^{3}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[4]{{(x - 4)}^{3}}}^{3}$ as $\sqrt[4]{{({(x - 4)}^{3})}^{3}}$ .

$\frac{2 \sqrt{x - 4} \sqrt[4]{{({(x - 4)}^{3})}^{3}}}{{(x - 4)}^{3}}$

Step 4.2

Multiply the exponents in ${({(x - 4)}^{3})}^{3}$ .

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Step 4.2.1

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

$\frac{2 \sqrt{x - 4} \sqrt[4]{{(x - 4)}^{3 \cdot 3}}}{{(x - 4)}^{3}}$

Step 4.2.2

Multiply $3$ by $3$ .

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

Step 4.3

Rewrite ${(x - 4)}^{9}$ as ${({(x - 4)}^{2})}^{4} (x - 4)$ .

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Step 4.3.1

Factor out ${(x - 4)}^{8}$ .

$\frac{2 \sqrt{x - 4} \sqrt[4]{{(x - 4)}^{8} (x - 4)}}{{(x - 4)}^{3}}$

Step 4.3.2

Rewrite ${(x - 4)}^{8}$ as ${({(x - 4)}^{2})}^{4}$ .

$\frac{2 \sqrt{x - 4} \sqrt[4]{{({(x - 4)}^{2})}^{4} (x - 4)}}{{(x - 4)}^{3}}$

Step 4.4

Pull terms out from under the radical.

$\frac{2 \sqrt{x - 4} ({(x - 4)}^{2} \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.5

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$\frac{2 \sqrt{x - 4} ((x - 4) (x - 4) \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.6

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 4.6.1

Apply the distributive property.

$\frac{2 \sqrt{x - 4} ((x (x - 4) - 4 (x - 4)) \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.6.2

Apply the distributive property.

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

Step 4.6.3

Apply the distributive property.

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

Step 4.7

Simplify and combine like terms.

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Step 4.7.1

Simplify each term.

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Step 4.7.1.1

Multiply $x$ by $x$ .

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

Step 4.7.1.2

Move $- 4$ to the left of $x$ .

$\frac{2 \sqrt{x - 4} ((x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.7.1.3

Multiply $- 4$ by $- 4$ .

$\frac{2 \sqrt{x - 4} ((x^{2} - 4 x - 4 x + 16) \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.7.2

Subtract $4 x$ from $- 4 x$ .

$\frac{2 \sqrt{x - 4} ((x^{2} - 8 x + 16) \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.8

Apply the distributive property.

$\frac{2 \sqrt{x - 4} (x^{2} \sqrt[4]{x - 4} - 8 x \sqrt[4]{x - 4} + 16 \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.9

Factor $\sqrt[4]{x - 4}$ out of $x^{2} \sqrt[4]{x - 4} - 8 x \sqrt[4]{x - 4} + 16 \sqrt[4]{x - 4}$ .

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Step 4.9.1

Factor $\sqrt[4]{x - 4}$ out of $x^{2} \sqrt[4]{x - 4}$ .

$\frac{2 \sqrt{x - 4} (\sqrt[4]{x - 4} x^{2} - 8 x \sqrt[4]{x - 4} + 16 \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.9.2

Factor $\sqrt[4]{x - 4}$ out of $- 8 x \sqrt[4]{x - 4}$ .

$\frac{2 \sqrt{x - 4} (\sqrt[4]{x - 4} x^{2} + \sqrt[4]{x - 4} (- 8 x) + 16 \sqrt[4]{x - 4})}{{(x - 4)}^{3}}$

Step 4.9.3

Factor $\sqrt[4]{x - 4}$ out of $16 \sqrt[4]{x - 4}$ .

$\frac{2 \sqrt{x - 4} (\sqrt[4]{x - 4} x^{2} + \sqrt[4]{x - 4} (- 8 x) + \sqrt[4]{x - 4} \cdot 16)}{{(x - 4)}^{3}}$

Step 4.9.4

Factor $\sqrt[4]{x - 4}$ out of $\sqrt[4]{x - 4} x^{2} + \sqrt[4]{x - 4} (- 8 x)$ .

$\frac{2 \sqrt{x - 4} (\sqrt[4]{x - 4} (x^{2} - 8 x) + \sqrt[4]{x - 4} \cdot 16)}{{(x - 4)}^{3}}$

Step 4.9.5

Factor $\sqrt[4]{x - 4}$ out of $\sqrt[4]{x - 4} (x^{2} - 8 x) + \sqrt[4]{x - 4} \cdot 16$ .

$\frac{2 \sqrt{x - 4} (\sqrt[4]{x - 4} (x^{2} - 8 x + 16))}{{(x - 4)}^{3}}$

Step 4.10

Factor using the perfect square rule.

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Step 4.10.1

Rewrite $16$ as $4^{2}$ .

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

Step 4.10.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 4.10.3

Rewrite the polynomial.

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

Step 4.10.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

$\frac{2 \sqrt{x - 4} (\sqrt[4]{x - 4} {(x - 4)}^{2})}{{(x - 4)}^{3}}$

$\frac{2 \sqrt{x - 4} \sqrt[4]{x - 4} {(x - 4)}^{2}}{{(x - 4)}^{3}}$

Step 4.11

Combine exponents.

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Step 4.11.1

Rewrite the expression using the least common index of $4$ .

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Step 4.11.1.1

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

$\frac{2 (\sqrt[4]{x - 4} {(x - 4)}^{\frac{1}{2}}) {(x - 4)}^{2}}{{(x - 4)}^{3}}$

Step 4.11.1.2

Rewrite ${(x - 4)}^{\frac{1}{2}}$ as ${(x - 4)}^{\frac{2}{4}}$ .

$\frac{2 (\sqrt[4]{x - 4} {(x - 4)}^{\frac{2}{4}}) {(x - 4)}^{2}}{{(x - 4)}^{3}}$

Step 4.11.1.3

Rewrite ${(x - 4)}^{\frac{2}{4}}$ as $\sqrt[4]{{(x - 4)}^{2}}$ .

$\frac{2 (\sqrt[4]{x - 4} \sqrt[4]{{(x - 4)}^{2}}) {(x - 4)}^{2}}{{(x - 4)}^{3}}$

Step 4.11.2

Combine using the product rule for radicals.

$\frac{2 \sqrt[4]{(x - 4) {(x - 4)}^{2}} {(x - 4)}^{2}}{{(x - 4)}^{3}}$

Step 4.11.3

Multiply $x - 4$ by ${(x - 4)}^{2}$ by adding the exponents.

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Step 4.11.3.1

Multiply $x - 4$ by ${(x - 4)}^{2}$ .

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Step 4.11.3.1.1

Raise $x - 4$ to the power of $1$ .

$\frac{2 \sqrt[4]{{(x - 4)}^{1} {(x - 4)}^{2}} {(x - 4)}^{2}}{{(x - 4)}^{3}}$

Step 4.11.3.1.2

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

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

Step 4.11.3.2

Add $1$ and $2$ .

$\frac{2 \sqrt[4]{{(x - 4)}^{3}} {(x - 4)}^{2}}{{(x - 4)}^{3}}$

Step 5

Cancel the common factor of ${(x - 4)}^{2}$ and ${(x - 4)}^{3}$ .

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Step 5.1

Factor ${(x - 4)}^{2}$ out of $2 \sqrt[4]{{(x - 4)}^{3}} {(x - 4)}^{2}$ .

$\frac{{(x - 4)}^{2} (2 \sqrt[4]{{(x - 4)}^{3}})}{{(x - 4)}^{3}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor ${(x - 4)}^{2}$ out of ${(x - 4)}^{3}$ .

$\frac{{(x - 4)}^{2} (2 \sqrt[4]{{(x - 4)}^{3}})}{{(x - 4)}^{2} (x - 4)}$

Step 5.2.2

Cancel the common factor.

$\frac{{(x - 4)}^{2} (2 \sqrt[4]{{(x - 4)}^{3}})}{{(x - 4)}^{2} (x - 4)}$

Step 5.2.3

Rewrite the expression.

$\frac{2 \sqrt[4]{{(x - 4)}^{3}}}{x - 4}$