Trigonometry Examples

$6 \sqrt{2} (1 + \sqrt{50})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 1.1.1

Factor $25$ out of $50$ .

$6 \sqrt{2} (1 + \sqrt{25 (2)})$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Simplify by multiplying through.

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Step 2.1

Apply the distributive property.

$6 \sqrt{2} \cdot 1 + 6 \sqrt{2} (5 \sqrt{2})$

Step 2.2

Multiply $6$ by $1$ .

$6 \sqrt{2} + 6 \sqrt{2} (5 \sqrt{2})$

Step 3

Multiply $6 \sqrt{2} (5 \sqrt{2})$ .

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Step 3.1

Multiply $5$ by $6$ .

$6 \sqrt{2} + 30 \sqrt{2} \sqrt{2}$

Step 3.2

Raise $\sqrt{2}$ to the power of $1$ .

$6 \sqrt{2} + 30 ({\sqrt{2}}^{1} \sqrt{2})$

Step 3.3

Raise $\sqrt{2}$ to the power of $1$ .

$6 \sqrt{2} + 30 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 3.4

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

$6 \sqrt{2} + 30 {\sqrt{2}}^{1 + 1}$

Step 3.5

Add $1$ and $1$ .

$6 \sqrt{2} + 30 {\sqrt{2}}^{2}$

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

$6 \sqrt{2} + 30 {(2^{\frac{1}{2}})}^{2}$

Step 4.1.2

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

$6 \sqrt{2} + 30 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 4.1.3

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

$6 \sqrt{2} + 30 \cdot 2^{\frac{2}{2}}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$6 \sqrt{2} + 30 \cdot 2^{\frac{2}{2}}$

Step 4.1.4.2

Rewrite the expression.

$6 \sqrt{2} + 30 \cdot 2^{1}$

Step 4.1.5

Evaluate the exponent.

$6 \sqrt{2} + 30 \cdot 2$

Step 4.2

Multiply $30$ by $2$ .

$6 \sqrt{2} + 60$

Step 5

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{2} + 60$

Decimal Form:

$68.48528137 \dots$