Trigonometry Examples

$v ({(- 2 \sqrt{5} - \sqrt{5})}^{2} + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1

Simplify each term.

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Step 1.1

Subtract $\sqrt{5}$ from $- 2 \sqrt{5}$ .

$v ({(- 3 \sqrt{5})}^{2} + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1.2

Apply the product rule to $- 3 \sqrt{5}$ .

$v ({(- 3)}^{2} {\sqrt{5}}^{2} + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1.3

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

$v (9 {\sqrt{5}}^{2} + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 1.4.2

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

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

Step 1.4.3

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

$v (9 \cdot 5^{\frac{2}{2}} + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1.4.4

Cancel the common factor of $2$ .

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Step 1.4.4.1

Cancel the common factor.

$v (9 \cdot 5^{\frac{2}{2}} + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1.4.4.2

Rewrite the expression.

$v (9 \cdot 5^{1} + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1.4.5

Evaluate the exponent.

$v (9 \cdot 5 + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1.5

Multiply $9$ by $5$ .

$v (45 + {(4 \sqrt{2} - \sqrt{2})}^{2})$

Step 1.6

Subtract $\sqrt{2}$ from $4 \sqrt{2}$ .

$v (45 + {(3 \sqrt{2})}^{2})$

Step 1.7

Apply the product rule to $3 \sqrt{2}$ .

$v (45 + 3^{2} {\sqrt{2}}^{2})$

Step 1.8

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

$v (45 + 9 {\sqrt{2}}^{2})$

Step 1.9

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

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Step 1.9.1

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

$v (45 + 9 {(2^{\frac{1}{2}})}^{2})$

Step 1.9.2

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

$v (45 + 9 \cdot 2^{\frac{1}{2} \cdot 2})$

Step 1.9.3

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

$v (45 + 9 \cdot 2^{\frac{2}{2}})$

Step 1.9.4

Cancel the common factor of $2$ .

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Step 1.9.4.1

Cancel the common factor.

$v (45 + 9 \cdot 2^{\frac{2}{2}})$

Step 1.9.4.2

Rewrite the expression.

$v (45 + 9 \cdot 2^{1})$

Step 1.9.5

Evaluate the exponent.

$v (45 + 9 \cdot 2)$

Step 1.10

Multiply $9$ by $2$ .

$v (45 + 18)$

Step 2

Simplify the expression.

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Step 2.1

Add $45$ and $18$ .

$v \cdot 63$

Step 2.2

Move $63$ to the left of $v$ .

$63 v$