Trigonometry Examples

$5 \sqrt{2 x^{3} x (\sqrt{10 x^{5} + 3 \sqrt{- 6 x^{2}}})}$

Step 1

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

$5 \sqrt{2 (x^{1} x^{3}) \sqrt{10 x^{5} + 3 \sqrt{- 6 x^{2}}}}$

Step 2

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

$5 \sqrt{2 x^{1 + 3} \sqrt{10 x^{5} + 3 \sqrt{- 6 x^{2}}}}$

Step 3

Add $1$ and $3$ .

$5 \sqrt{2 x^{4} \sqrt{10 x^{5} + 3 \sqrt{- 6 x^{2}}}}$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 6 x^{2}}$ as ${(- 6 x^{2})}^{\frac{1}{2}}$ .

$5 \sqrt{2 x^{4} \sqrt{10 x^{5} + 3 {(- 6 x^{2})}^{\frac{1}{2}}}}$

Step 5

Apply the product rule to $- 6 x^{2}$ .

$5 \sqrt{2 x^{4} \sqrt{10 x^{5} + 3 ({(- 6)}^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}})}}$

Step 6

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

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Step 6.1

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

$5 \sqrt{2 x^{4} \sqrt{10 x^{5} + 3 ({(- 6)}^{\frac{1}{2}} x^{2 (\frac{1}{2})})}}$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Cancel the common factor.

$5 \sqrt{2 x^{4} \sqrt{10 x^{5} + 3 ({(- 6)}^{\frac{1}{2}} x^{2 (\frac{1}{2})})}}$

Step 6.2.2

Rewrite the expression.

$5 \sqrt{2 x^{4} \sqrt{10 x^{5} + 3 ({(- 6)}^{\frac{1}{2}} x^{1})}}$

Step 7

Simplify.

$5 \sqrt{2 x^{4} \sqrt{10 x^{5} + 3 ({(- 6)}^{\frac{1}{2}} x)}}$

Step 8

Factor $x$ out of $10 x^{5} + 3 {(- 6)}^{\frac{1}{2}} x$ .

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Step 8.1

Factor $x$ out of $10 x^{5}$ .

$5 \sqrt{2 x^{4} \sqrt{x (10 x^{4}) + 3 {(- 6)}^{\frac{1}{2}} x}}$

Step 8.2

Factor $x$ out of $3 {(- 6)}^{\frac{1}{2}} x$ .

$5 \sqrt{2 x^{4} \sqrt{x (10 x^{4}) + x (3 {(- 6)}^{\frac{1}{2}})}}$

Step 8.3

Factor $x$ out of $x (10 x^{4}) + x (3 {(- 6)}^{\frac{1}{2}})$ .

$5 \sqrt{2 x^{4} \sqrt{x (10 x^{4} + 3 {(- 6)}^{\frac{1}{2}})}}$

Step 9

Rewrite $2 x^{4} \sqrt{x (10 x^{4} + 3 {(- 6)}^{\frac{1}{2}})}$ as ${(x^{2})}^{2} \cdot (2 \sqrt{x (10 {(x^{2})}^{2} + 3 {(- 6)}^{\frac{1}{2}})})$ .

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Step 9.1

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

$5 \sqrt{2 {(x^{2})}^{2} \sqrt{x (10 x^{4} + 3 {(- 6)}^{\frac{1}{2}})}}$

Step 9.2

Reorder $2$ and ${(x^{2})}^{2}$ .

$5 \sqrt{{(x^{2})}^{2} \cdot 2 \sqrt{x (10 x^{4} + 3 {(- 6)}^{\frac{1}{2}})}}$

Step 9.3

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

$5 \sqrt{{(x^{2})}^{2} \cdot 2 \sqrt{x (10 {(x^{2})}^{2} + 3 {(- 6)}^{\frac{1}{2}})}}$

Step 9.4

Add parentheses.

$5 \sqrt{{(x^{2})}^{2} \cdot (2 \sqrt{x (10 {(x^{2})}^{2} + 3 {(- 6)}^{\frac{1}{2}})})}$

Step 10

Pull terms out from under the radical.

$5 (x^{2} \sqrt{2 \sqrt{x (10 {(x^{2})}^{2} + 3 {(- 6)}^{\frac{1}{2}})}})$

Step 11

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

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Step 11.1

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

$5 (x^{2} \sqrt{2 \sqrt{x (10 x^{2 \cdot 2} + 3 {(- 6)}^{\frac{1}{2}})}})$

Step 11.2

Multiply $2$ by $2$ .

$5 (x^{2} \sqrt{2 \sqrt{x (10 x^{4} + 3 {(- 6)}^{\frac{1}{2}})}})$

$5 x^{2} \sqrt{2 \sqrt{x (10 x^{4} + 3 {(- 6)}^{\frac{1}{2}})}}$