Trigonometry Examples

$2 (\frac{\sqrt{80}}{9}) (\frac{\sqrt{80}}{9})$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 1.1.1

Factor $16$ out of $80$ .

$2 (\frac{\sqrt{80}}{9}) \cdot \frac{\sqrt{16 (5)}}{9}$

Step 1.1.2

Rewrite $16$ as $4^{2}$ .

$2 (\frac{\sqrt{80}}{9}) \cdot \frac{\sqrt{4^{2} \cdot 5}}{9}$

Step 1.2

Pull terms out from under the radical.

$2 (\frac{\sqrt{80}}{9}) \cdot \frac{4 \sqrt{5}}{9}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 2.1.1

Factor $16$ out of $80$ .

$2 \frac{\sqrt{16 (5)}}{9} \cdot \frac{4 \sqrt{5}}{9}$

Step 2.1.2

Rewrite $16$ as $4^{2}$ .

$2 \frac{\sqrt{4^{2} \cdot 5}}{9} \cdot \frac{4 \sqrt{5}}{9}$

Step 2.2

Pull terms out from under the radical.

$2 \frac{4 \sqrt{5}}{9} \cdot \frac{4 \sqrt{5}}{9}$

Step 3

Multiply $2 \frac{4 \sqrt{5}}{9}$ .

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Step 3.1

Combine $2$ and $\frac{4 \sqrt{5}}{9}$ .

$\frac{2 (4 \sqrt{5})}{9} \cdot \frac{4 \sqrt{5}}{9}$

Step 3.2

Multiply $4$ by $2$ .

$\frac{8 \sqrt{5}}{9} \cdot \frac{4 \sqrt{5}}{9}$

Step 4

Multiply $\frac{8 \sqrt{5}}{9} \cdot \frac{4 \sqrt{5}}{9}$ .

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Step 4.1

Multiply $\frac{8 \sqrt{5}}{9}$ by $\frac{4 \sqrt{5}}{9}$ .

$\frac{8 \sqrt{5} (4 \sqrt{5})}{9 \cdot 9}$

Step 4.2

Multiply $4$ by $8$ .

$\frac{32 \sqrt{5} \sqrt{5}}{9 \cdot 9}$

Step 4.3

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

$\frac{32 ({\sqrt{5}}^{1} \sqrt{5})}{9 \cdot 9}$

Step 4.4

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

$\frac{32 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{9 \cdot 9}$

Step 4.5

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

$\frac{32 {\sqrt{5}}^{1 + 1}}{9 \cdot 9}$

Step 4.6

Add $1$ and $1$ .

$\frac{32 {\sqrt{5}}^{2}}{9 \cdot 9}$

Step 4.7

Multiply $9$ by $9$ .

$\frac{32 {\sqrt{5}}^{2}}{81}$

Step 5

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

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Step 5.1

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

$\frac{32 {(5^{\frac{1}{2}})}^{2}}{81}$

Step 5.2

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

$\frac{32 \cdot 5^{\frac{1}{2} \cdot 2}}{81}$

Step 5.3

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

$\frac{32 \cdot 5^{\frac{2}{2}}}{81}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\frac{32 \cdot 5^{\frac{2}{2}}}{81}$

Step 5.4.2

Rewrite the expression.

$\frac{32 \cdot 5^{1}}{81}$

Step 5.5

Evaluate the exponent.

$\frac{32 \cdot 5}{81}$

Step 6

Multiply $32$ by $5$ .

$\frac{160}{81}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{160}{81}$

Decimal Form:

$1.97530864 \dots$

Mixed Number Form:

$1 \frac{79}{81}$