Trigonometry Examples

$5 \sqrt{6} (3 \sqrt{6} - 6 \sqrt{3})$

Step 1

Apply the distributive property.

$5 \sqrt{6} (3 \sqrt{6}) + 5 \sqrt{6} (- 6 \sqrt{3})$

Step 2

Multiply $5 \sqrt{6} (3 \sqrt{6})$ .

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Step 2.1

Multiply $3$ by $5$ .

$15 \sqrt{6} \sqrt{6} + 5 \sqrt{6} (- 6 \sqrt{3})$

Step 2.2

Raise $\sqrt{6}$ to the power of $1$ .

$15 ({\sqrt{6}}^{1} \sqrt{6}) + 5 \sqrt{6} (- 6 \sqrt{3})$

Step 2.3

Raise $\sqrt{6}$ to the power of $1$ .

$15 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) + 5 \sqrt{6} (- 6 \sqrt{3})$

Step 2.4

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

$15 {\sqrt{6}}^{1 + 1} + 5 \sqrt{6} (- 6 \sqrt{3})$

Step 2.5

Add $1$ and $1$ .

$15 {\sqrt{6}}^{2} + 5 \sqrt{6} (- 6 \sqrt{3})$

Step 3

Multiply $5 \sqrt{6} (- 6 \sqrt{3})$ .

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Step 3.1

Multiply $- 6$ by $5$ .

$15 {\sqrt{6}}^{2} - 30 \sqrt{6} \sqrt{3}$

Step 3.2

Combine using the product rule for radicals.

$15 {\sqrt{6}}^{2} - 30 \sqrt{3 \cdot 6}$

Step 3.3

Multiply $3$ by $6$ .

$15 {\sqrt{6}}^{2} - 30 \sqrt{18}$

Step 4

Simplify each term.

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Step 4.1

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.1.1

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

$15 {(6^{\frac{1}{2}})}^{2} - 30 \sqrt{18}$

Step 4.1.2

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

$15 \cdot 6^{\frac{1}{2} \cdot 2} - 30 \sqrt{18}$

Step 4.1.3

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

$15 \cdot 6^{\frac{2}{2}} - 30 \sqrt{18}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$15 \cdot 6^{\frac{2}{2}} - 30 \sqrt{18}$

Step 4.1.4.2

Rewrite the expression.

$15 \cdot 6^{1} - 30 \sqrt{18}$

Step 4.1.5

Evaluate the exponent.

$15 \cdot 6 - 30 \sqrt{18}$

Step 4.2

Multiply $15$ by $6$ .

$90 - 30 \sqrt{18}$

Step 4.3

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.3.1

Factor $9$ out of $18$ .

$90 - 30 \sqrt{9 (2)}$

Step 4.3.2

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

$90 - 30 \sqrt{3^{2} \cdot 2}$

Step 4.4

Pull terms out from under the radical.

$90 - 30 (3 \sqrt{2})$

Step 4.5

Multiply $3$ by $- 30$ .

$90 - 90 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$90 - 90 \sqrt{2}$

Decimal Form:

$- 37.27922061 \dots$