Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the least common index of $6$ .

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(a + x)}^{3}}$ as ${(a + x)}^{\frac{3}{2}}$ .

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

Step 1.1.2

Rewrite ${(a + x)}^{\frac{3}{2}}$ as ${(a + x)}^{\frac{9}{6}}$ .

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

Step 1.1.3

Rewrite ${(a + x)}^{\frac{9}{6}}$ as $\sqrt[6]{{(a + x)}^{9}}$ .

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

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(a + x)}^{2}}$ as ${(a + x)}^{\frac{2}{3}}$ .

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

Step 1.1.5

Rewrite ${(a + x)}^{\frac{2}{3}}$ as ${(a + x)}^{\frac{4}{6}}$ .

$\frac{\sqrt[6]{{(a + x)}^{9}} {(a + x)}^{\frac{4}{6}}}{\sqrt[4]{a + x}}$

Step 1.1.6

Rewrite ${(a + x)}^{\frac{4}{6}}$ as $\sqrt[6]{{(a + x)}^{4}}$ .

$\frac{\sqrt[6]{{(a + x)}^{9}} \sqrt[6]{{(a + x)}^{4}}}{\sqrt[4]{a + x}}$

Step 1.2

Combine using the product rule for radicals.

$\frac{\sqrt[6]{{(a + x)}^{9} {(a + x)}^{4}}}{\sqrt[4]{a + x}}$

Step 1.3

Multiply ${(a + x)}^{9}$ by ${(a + x)}^{4}$ by adding the exponents.

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Step 1.3.1

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

$\frac{\sqrt[6]{{(a + x)}^{9 + 4}}}{\sqrt[4]{a + x}}$

Step 1.3.2

Add $9$ and $4$ .

$\frac{\sqrt[6]{{(a + x)}^{13}}}{\sqrt[4]{a + x}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite ${(a + x)}^{13}$ as ${({(a + x)}^{2})}^{6} (a + x)$ .

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Step 2.1.1

Factor out ${(a + x)}^{12}$ .

$\frac{\sqrt[6]{{(a + x)}^{12} (a + x)}}{\sqrt[4]{a + x}}$

Step 2.1.2

Rewrite ${(a + x)}^{12}$ as ${({(a + x)}^{2})}^{6}$ .

$\frac{\sqrt[6]{{({(a + x)}^{2})}^{6} (a + x)}}{\sqrt[4]{a + x}}$

Step 2.2

Pull terms out from under the radical.

$\frac{{(a + x)}^{2} \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.3

Rewrite ${(a + x)}^{2}$ as $(a + x) (a + x)$ .

$\frac{(a + x) (a + x) \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.4

Expand $(a + x) (a + x)$ using the FOIL Method.

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Step 2.4.1

Apply the distributive property.

$\frac{(a (a + x) + x (a + x)) \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.4.2

Apply the distributive property.

$\frac{(a \cdot a + a x + x (a + x)) \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.4.3

Apply the distributive property.

$\frac{(a \cdot a + a x + x a + x \cdot x) \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.5

Simplify and combine like terms.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Multiply $a$ by $a$ .

$\frac{(a^{2} + a x + x a + x \cdot x) \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.5.1.2

Multiply $x$ by $x$ .

$\frac{(a^{2} + a x + x a + x^{2}) \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.5.2

Add $a x$ and $x a$ .

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Step 2.5.2.1

Reorder $x$ and $a$ .

$\frac{(a^{2} + a x + a x + x^{2}) \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.5.2.2

Add $a x$ and $a x$ .

$\frac{(a^{2} + 2 a x + x^{2}) \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.6

Apply the distributive property.

$\frac{a^{2} \sqrt[6]{a + x} + 2 a x \sqrt[6]{a + x} + x^{2} \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.7

Factor $\sqrt[6]{a + x}$ out of $a^{2} \sqrt[6]{a + x} + 2 a x \sqrt[6]{a + x} + x^{2} \sqrt[6]{a + x}$ .

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Step 2.7.1

Factor $\sqrt[6]{a + x}$ out of $a^{2} \sqrt[6]{a + x}$ .

$\frac{\sqrt[6]{a + x} a^{2} + 2 a x \sqrt[6]{a + x} + x^{2} \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.7.2

Factor $\sqrt[6]{a + x}$ out of $2 a x \sqrt[6]{a + x}$ .

$\frac{\sqrt[6]{a + x} a^{2} + \sqrt[6]{a + x} (2 a x) + x^{2} \sqrt[6]{a + x}}{\sqrt[4]{a + x}}$

Step 2.7.3

Factor $\sqrt[6]{a + x}$ out of $x^{2} \sqrt[6]{a + x}$ .

$\frac{\sqrt[6]{a + x} a^{2} + \sqrt[6]{a + x} (2 a x) + \sqrt[6]{a + x} x^{2}}{\sqrt[4]{a + x}}$

Step 2.7.4

Factor $\sqrt[6]{a + x}$ out of $\sqrt[6]{a + x} a^{2} + \sqrt[6]{a + x} (2 a x)$ .

$\frac{\sqrt[6]{a + x} (a^{2} + 2 a x) + \sqrt[6]{a + x} x^{2}}{\sqrt[4]{a + x}}$

Step 2.7.5

Factor $\sqrt[6]{a + x}$ out of $\sqrt[6]{a + x} (a^{2} + 2 a x) + \sqrt[6]{a + x} x^{2}$ .

$\frac{\sqrt[6]{a + x} (a^{2} + 2 a x + x^{2})}{\sqrt[4]{a + x}}$

Step 2.8

Factor using the perfect square rule.

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Step 2.8.1

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

$2 a x = 2 \cdot a \cdot x$

Step 2.8.2

Rewrite the polynomial.

$\frac{\sqrt[6]{a + x} (a^{2} + 2 \cdot a \cdot x + x^{2})}{\sqrt[4]{a + x}}$

Step 2.8.3

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

$\frac{\sqrt[6]{a + x} {(a + x)}^{2}}{\sqrt[4]{a + x}}$

Step 3

Multiply $\frac{\sqrt[6]{a + x} {(a + x)}^{2}}{\sqrt[4]{a + x}}$ by $\frac{{\sqrt[4]{a + x}}^{3}}{{\sqrt[4]{a + x}}^{3}}$ .

$\frac{\sqrt[6]{a + x} {(a + x)}^{2}}{\sqrt[4]{a + x}} \cdot \frac{{\sqrt[4]{a + x}}^{3}}{{\sqrt[4]{a + x}}^{3}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt[6]{a + x} {(a + x)}^{2}}{\sqrt[4]{a + x}}$ by $\frac{{\sqrt[4]{a + x}}^{3}}{{\sqrt[4]{a + x}}^{3}}$ .

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{\sqrt[4]{a + x} {\sqrt[4]{a + x}}^{3}}$

Step 4.2

Raise $\sqrt[4]{a + x}$ to the power of $1$ .

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{{\sqrt[4]{a + x}}^{1} {\sqrt[4]{a + x}}^{3}}$

Step 4.3

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

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{{\sqrt[4]{a + x}}^{1 + 3}}$

Step 4.4

Add $1$ and $3$ .

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{{\sqrt[4]{a + x}}^{4}}$

Step 4.5

Rewrite ${\sqrt[4]{a + x}}^{4}$ as $a + x$ .

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Step 4.5.1

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

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{{({(a + x)}^{\frac{1}{4}})}^{4}}$

Step 4.5.2

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

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{{(a + x)}^{\frac{1}{4} \cdot 4}}$

Step 4.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{{(a + x)}^{\frac{4}{4}}}$

Step 4.5.4

Cancel the common factor of $4$ .

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Step 4.5.4.1

Cancel the common factor.

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{{(a + x)}^{\frac{4}{4}}}$

Step 4.5.4.2

Rewrite the expression.

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{{(a + x)}^{1}}$

Step 4.5.5

Simplify.

$\frac{\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}}{a + x}$

Step 5

Cancel the common factor of ${(a + x)}^{2}$ and $a + x$ .

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Step 5.1

Factor $a + x$ out of $\sqrt[6]{a + x} {(a + x)}^{2} {\sqrt[4]{a + x}}^{3}$ .

$\frac{(a + x) (\sqrt[6]{a + x} (a + x) {\sqrt[4]{a + x}}^{3})}{a + x}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Multiply by $1$ .

$\frac{(a + x) (\sqrt[6]{a + x} (a + x) {\sqrt[4]{a + x}}^{3})}{(a + x) \cdot 1}$

Step 5.2.2

Cancel the common factor.

$\frac{(a + x) (\sqrt[6]{a + x} (a + x) {\sqrt[4]{a + x}}^{3})}{(a + x) \cdot 1}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt[6]{a + x} (a + x) {\sqrt[4]{a + x}}^{3}}{1}$

Step 5.2.4

Divide $\sqrt[6]{a + x} (a + x) {\sqrt[4]{a + x}}^{3}$ by $1$ .

$\sqrt[6]{a + x} (a + x) {\sqrt[4]{a + x}}^{3}$

Step 6

Apply the distributive property.

$(\sqrt[6]{a + x} a + \sqrt[6]{a + x} x) {\sqrt[4]{a + x}}^{3}$

Step 7

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

$(\sqrt[6]{a + x} a + \sqrt[6]{a + x} x) \sqrt[4]{{(a + x)}^{3}}$

Step 8

Apply the distributive property.

$\sqrt[6]{a + x} a \sqrt[4]{{(a + x)}^{3}} + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9

Multiply $\sqrt[6]{a + x} a \sqrt[4]{{(a + x)}^{3}}$ .

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Step 9.1

Rewrite the expression using the least common index of $12$ .

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Step 9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{a + x}$ as ${(a + x)}^{\frac{1}{6}}$ .

$a ({(a + x)}^{\frac{1}{6}} \sqrt[4]{{(a + x)}^{3}}) + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9.1.2

Rewrite ${(a + x)}^{\frac{1}{6}}$ as ${(a + x)}^{\frac{2}{12}}$ .

$a ({(a + x)}^{\frac{2}{12}} \sqrt[4]{{(a + x)}^{3}}) + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9.1.3

Rewrite ${(a + x)}^{\frac{2}{12}}$ as $\sqrt[12]{{(a + x)}^{2}}$ .

$a (\sqrt[12]{{(a + x)}^{2}} \sqrt[4]{{(a + x)}^{3}}) + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9.1.4

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

$a (\sqrt[12]{{(a + x)}^{2}} {(a + x)}^{\frac{3}{4}}) + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9.1.5

Rewrite ${(a + x)}^{\frac{3}{4}}$ as ${(a + x)}^{\frac{9}{12}}$ .

$a (\sqrt[12]{{(a + x)}^{2}} {(a + x)}^{\frac{9}{12}}) + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9.1.6

Rewrite ${(a + x)}^{\frac{9}{12}}$ as $\sqrt[12]{{(a + x)}^{9}}$ .

$a (\sqrt[12]{{(a + x)}^{2}} \sqrt[12]{{(a + x)}^{9}}) + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9.2

Combine using the product rule for radicals.

$a \sqrt[12]{{(a + x)}^{2} {(a + x)}^{9}} + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9.3

Multiply ${(a + x)}^{2}$ by ${(a + x)}^{9}$ by adding the exponents.

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Step 9.3.1

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

$a \sqrt[12]{{(a + x)}^{2 + 9}} + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 9.3.2

Add $2$ and $9$ .

$a \sqrt[12]{{(a + x)}^{11}} + \sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$

Step 10

Multiply $\sqrt[6]{a + x} x \sqrt[4]{{(a + x)}^{3}}$ .

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Step 10.1

Rewrite the expression using the least common index of $12$ .

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Step 10.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{a + x}$ as ${(a + x)}^{\frac{1}{6}}$ .

$a \sqrt[12]{{(a + x)}^{11}} + x ({(a + x)}^{\frac{1}{6}} \sqrt[4]{{(a + x)}^{3}})$

Step 10.1.2

Rewrite ${(a + x)}^{\frac{1}{6}}$ as ${(a + x)}^{\frac{2}{12}}$ .

$a \sqrt[12]{{(a + x)}^{11}} + x ({(a + x)}^{\frac{2}{12}} \sqrt[4]{{(a + x)}^{3}})$

Step 10.1.3

Rewrite ${(a + x)}^{\frac{2}{12}}$ as $\sqrt[12]{{(a + x)}^{2}}$ .

$a \sqrt[12]{{(a + x)}^{11}} + x (\sqrt[12]{{(a + x)}^{2}} \sqrt[4]{{(a + x)}^{3}})$

Step 10.1.4

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

$a \sqrt[12]{{(a + x)}^{11}} + x (\sqrt[12]{{(a + x)}^{2}} {(a + x)}^{\frac{3}{4}})$

Step 10.1.5

Rewrite ${(a + x)}^{\frac{3}{4}}$ as ${(a + x)}^{\frac{9}{12}}$ .

$a \sqrt[12]{{(a + x)}^{11}} + x (\sqrt[12]{{(a + x)}^{2}} {(a + x)}^{\frac{9}{12}})$

Step 10.1.6

Rewrite ${(a + x)}^{\frac{9}{12}}$ as $\sqrt[12]{{(a + x)}^{9}}$ .

$a \sqrt[12]{{(a + x)}^{11}} + x (\sqrt[12]{{(a + x)}^{2}} \sqrt[12]{{(a + x)}^{9}})$

Step 10.2

Combine using the product rule for radicals.

$a \sqrt[12]{{(a + x)}^{11}} + x \sqrt[12]{{(a + x)}^{2} {(a + x)}^{9}}$

Step 10.3

Multiply ${(a + x)}^{2}$ by ${(a + x)}^{9}$ by adding the exponents.

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Step 10.3.1

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

$a \sqrt[12]{{(a + x)}^{11}} + x \sqrt[12]{{(a + x)}^{2 + 9}}$

Step 10.3.2

Add $2$ and $9$ .

$a \sqrt[12]{{(a + x)}^{11}} + x \sqrt[12]{{(a + x)}^{11}}$