Trigonometry Examples

$2 \sqrt{5} (\sqrt{20} + 3)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 1.1.1

Factor $4$ out of $20$ .

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

Step 1.1.2

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

$2 \sqrt{5} (\sqrt{2^{2} \cdot 5} + 3)$

Step 1.2

Pull terms out from under the radical.

$2 \sqrt{5} (2 \sqrt{5} + 3)$

Step 2

Apply the distributive property.

$2 \sqrt{5} (2 \sqrt{5}) + 2 \sqrt{5} \cdot 3$

Step 3

Multiply $2 \sqrt{5} (2 \sqrt{5})$ .

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Step 3.1

Multiply $2$ by $2$ .

$4 \sqrt{5} \sqrt{5} + 2 \sqrt{5} \cdot 3$

Step 3.2

Raise $\sqrt{5}$ to the power of $1$ .

$4 ({\sqrt{5}}^{1} \sqrt{5}) + 2 \sqrt{5} \cdot 3$

Step 3.3

Raise $\sqrt{5}$ to the power of $1$ .

$4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 2 \sqrt{5} \cdot 3$

Step 3.4

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

$4 {\sqrt{5}}^{1 + 1} + 2 \sqrt{5} \cdot 3$

Step 3.5

Add $1$ and $1$ .

$4 {\sqrt{5}}^{2} + 2 \sqrt{5} \cdot 3$

Step 4

Multiply $3$ by $2$ .

$4 {\sqrt{5}}^{2} + 6 \sqrt{5}$

Step 5

Simplify each term.

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Step 5.1

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

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Step 5.1.1

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

$4 {(5^{\frac{1}{2}})}^{2} + 6 \sqrt{5}$

Step 5.1.2

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

$4 \cdot 5^{\frac{1}{2} \cdot 2} + 6 \sqrt{5}$

Step 5.1.3

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

$4 \cdot 5^{\frac{2}{2}} + 6 \sqrt{5}$

Step 5.1.4

Cancel the common factor of $2$ .

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Step 5.1.4.1

Cancel the common factor.

$4 \cdot 5^{\frac{2}{2}} + 6 \sqrt{5}$

Step 5.1.4.2

Rewrite the expression.

$4 \cdot 5^{1} + 6 \sqrt{5}$

Step 5.1.5

Evaluate the exponent.

$4 \cdot 5 + 6 \sqrt{5}$

Step 5.2

Multiply $4$ by $5$ .

$20 + 6 \sqrt{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$20 + 6 \sqrt{5}$

Decimal Form:

$33.41640786 \dots$