Trigonometry Examples

$4 \sqrt{\frac{150}{\sqrt{189 x}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $189 x$ as $3^{2} \cdot (21 x)$ .

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Step 1.1.1

Factor $9$ out of $189$ .

$4 \sqrt{\frac{150}{\sqrt{9 (21) x}}}$

Step 1.1.2

Rewrite $9$ as $3^{2}$ .

$4 \sqrt{\frac{150}{\sqrt{3^{2} \cdot 21 x}}}$

Step 1.1.3

Add parentheses.

$4 \sqrt{\frac{150}{\sqrt{3^{2} \cdot (21 x)}}}$

Step 1.2

Pull terms out from under the radical.

$4 \sqrt{\frac{150}{3 \sqrt{21 x}}}$

Step 2

Cancel the common factor of $150$ and $3$ .

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Step 2.1

Factor $3$ out of $150$ .

$4 \sqrt{\frac{3 \cdot 50}{3 \sqrt{21 x}}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $3 \sqrt{21 x}$ .

$4 \sqrt{\frac{3 \cdot 50}{3 (\sqrt{21 x})}}$

Step 2.2.2

Cancel the common factor.

$4 \sqrt{\frac{3 \cdot 50}{3 \sqrt{21 x}}}$

Step 2.2.3

Rewrite the expression.

$4 \sqrt{\frac{50}{\sqrt{21 x}}}$

Step 3

Multiply $\frac{50}{\sqrt{21 x}}$ by $\frac{\sqrt{21 x}}{\sqrt{21 x}}$ .

$4 \sqrt{\frac{50}{\sqrt{21 x}} \cdot \frac{\sqrt{21 x}}{\sqrt{21 x}}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{50}{\sqrt{21 x}}$ by $\frac{\sqrt{21 x}}{\sqrt{21 x}}$ .

$4 \sqrt{\frac{50 \sqrt{21 x}}{\sqrt{21 x} \sqrt{21 x}}}$

Step 4.2

Raise $\sqrt{21 x}$ to the power of $1$ .

$4 \sqrt{\frac{50 \sqrt{21 x}}{{\sqrt{21 x}}^{1} \sqrt{21 x}}}$

Step 4.3

Raise $\sqrt{21 x}$ to the power of $1$ .

$4 \sqrt{\frac{50 \sqrt{21 x}}{{\sqrt{21 x}}^{1} {\sqrt{21 x}}^{1}}}$

Step 4.4

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

$4 \sqrt{\frac{50 \sqrt{21 x}}{{\sqrt{21 x}}^{1 + 1}}}$

Step 4.5

Add $1$ and $1$ .

$4 \sqrt{\frac{50 \sqrt{21 x}}{{\sqrt{21 x}}^{2}}}$

Step 4.6

Rewrite ${\sqrt{21 x}}^{2}$ as $21 x$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21 x}$ as ${(21 x)}^{\frac{1}{2}}$ .

$4 \sqrt{\frac{50 \sqrt{21 x}}{{({(21 x)}^{\frac{1}{2}})}^{2}}}$

Step 4.6.2

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

$4 \sqrt{\frac{50 \sqrt{21 x}}{{(21 x)}^{\frac{1}{2} \cdot 2}}}$

Step 4.6.3

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

$4 \sqrt{\frac{50 \sqrt{21 x}}{{(21 x)}^{\frac{2}{2}}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$4 \sqrt{\frac{50 \sqrt{21 x}}{{(21 x)}^{\frac{2}{2}}}}$

Step 4.6.4.2

Rewrite the expression.

$4 \sqrt{\frac{50 \sqrt{21 x}}{{(21 x)}^{1}}}$

Step 4.6.5

Simplify.

$4 \sqrt{\frac{50 \sqrt{21 x}}{21 x}}$

Step 5

Rewrite $\sqrt{\frac{50 \sqrt{21 x}}{21 x}}$ as $\frac{\sqrt{50 \sqrt{21 x}}}{\sqrt{21 x}}$ .

$4 \frac{\sqrt{50 \sqrt{21 x}}}{\sqrt{21 x}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $50 \sqrt{21 x}$ as $5^{2} \cdot (2 \sqrt{21 x})$ .

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Step 6.1.1

Factor $25$ out of $50$ .

$4 \frac{\sqrt{25 (2) \sqrt{21 x}}}{\sqrt{21 x}}$

Step 6.1.2

Rewrite $25$ as $5^{2}$ .

$4 \frac{\sqrt{5^{2} \cdot 2 \sqrt{21 x}}}{\sqrt{21 x}}$

Step 6.1.3

Add parentheses.

$4 \frac{\sqrt{5^{2} \cdot (2 \sqrt{21 x})}}{\sqrt{21 x}}$

Step 6.2

Pull terms out from under the radical.

$4 \frac{5 \sqrt{2 \sqrt{21 x}}}{\sqrt{21 x}}$

Step 7

Multiply $\frac{5 \sqrt{2 \sqrt{21 x}}}{\sqrt{21 x}}$ by $\frac{\sqrt{21 x}}{\sqrt{21 x}}$ .

$4 (\frac{5 \sqrt{2 \sqrt{21 x}}}{\sqrt{21 x}} \cdot \frac{\sqrt{21 x}}{\sqrt{21 x}})$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{5 \sqrt{2 \sqrt{21 x}}}{\sqrt{21 x}}$ by $\frac{\sqrt{21 x}}{\sqrt{21 x}}$ .

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{\sqrt{21 x} \sqrt{21 x}}$

Step 8.2

Raise $\sqrt{21 x}$ to the power of $1$ .

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{\sqrt{21 x}}^{1} \sqrt{21 x}}$

Step 8.3

Raise $\sqrt{21 x}$ to the power of $1$ .

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{\sqrt{21 x}}^{1} {\sqrt{21 x}}^{1}}$

Step 8.4

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

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{\sqrt{21 x}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{\sqrt{21 x}}^{2}}$

Step 8.6

Rewrite ${\sqrt{21 x}}^{2}$ as $21 x$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21 x}$ as ${(21 x)}^{\frac{1}{2}}$ .

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{({(21 x)}^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{(21 x)}^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{(21 x)}^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{(21 x)}^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{{(21 x)}^{1}}$

Step 8.6.5

Simplify.

$4 \frac{5 \sqrt{2 \sqrt{21 x}} \sqrt{21 x}}{21 x}$

Step 9

Simplify the numerator.

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Step 9.1

Combine using the product rule for radicals.

$4 \frac{5 \sqrt{21 x (2 \sqrt{21 x})}}{21 x}$

Step 9.2

Multiply $2$ by $21$ .

$4 \frac{5 \sqrt{42 x \sqrt{21 x}}}{21 x}$

Step 10

Multiply $4 \frac{5 \sqrt{42 x \sqrt{21 x}}}{21 x}$ .

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Step 10.1

Combine $4$ and $\frac{5 \sqrt{42 x \sqrt{21 x}}}{21 x}$ .

$\frac{4 (5 \sqrt{42 x \sqrt{21 x}})}{21 x}$

Step 10.2

Multiply $5$ by $4$ .

$\frac{20 \sqrt{42 x \sqrt{21 x}}}{21 x}$