Trigonometry Examples

$(5 \sqrt{8} + 5) (2 \sqrt{8} - 2)$

Step 1

Simplify terms.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 1.1.1.1

Factor $4$ out of $8$ .

$(5 \sqrt{4 (2)} + 5) (2 \sqrt{8} - 2)$

Step 1.1.1.2

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

$(5 \sqrt{2^{2} \cdot 2} + 5) (2 \sqrt{8} - 2)$

Step 1.1.2

Pull terms out from under the radical.

$(5 (2 \sqrt{2}) + 5) (2 \sqrt{8} - 2)$

Step 1.1.3

Multiply $2$ by $5$ .

$(10 \sqrt{2} + 5) (2 \sqrt{8} - 2)$

Step 1.2

Simplify each term.

Tap for more steps...

Step 1.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 1.2.1.1

Factor $4$ out of $8$ .

$(10 \sqrt{2} + 5) (2 \sqrt{4 (2)} - 2)$

Step 1.2.1.2

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

$(10 \sqrt{2} + 5) (2 \sqrt{2^{2} \cdot 2} - 2)$

Step 1.2.2

Pull terms out from under the radical.

$(10 \sqrt{2} + 5) (2 (2 \sqrt{2}) - 2)$

Step 1.2.3

Multiply $2$ by $2$ .

$(10 \sqrt{2} + 5) (4 \sqrt{2} - 2)$

Step 2

Expand $(10 \sqrt{2} + 5) (4 \sqrt{2} - 2)$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

$10 \sqrt{2} (4 \sqrt{2} - 2) + 5 (4 \sqrt{2} - 2)$

Step 2.2

Apply the distributive property.

$10 \sqrt{2} (4 \sqrt{2}) + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2} - 2)$

Step 2.3

Apply the distributive property.

$10 \sqrt{2} (4 \sqrt{2}) + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Multiply $10 \sqrt{2} (4 \sqrt{2})$ .

Tap for more steps...

Step 3.1.1.1

Multiply $4$ by $10$ .

$40 \sqrt{2} \sqrt{2} + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.1.2

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

$40 ({\sqrt{2}}^{1} \sqrt{2}) + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.1.3

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

$40 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.1.4

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

$40 {\sqrt{2}}^{1 + 1} + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.1.5

Add $1$ and $1$ .

$40 {\sqrt{2}}^{2} + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.2

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

Tap for more steps...

Step 3.1.2.1

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

$40 {(2^{\frac{1}{2}})}^{2} + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.2.2

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

$40 \cdot 2^{\frac{1}{2} \cdot 2} + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.2.3

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

$40 \cdot 2^{\frac{2}{2}} + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.2.4.1

Cancel the common factor.

$40 \cdot 2^{\frac{2}{2}} + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.2.4.2

Rewrite the expression.

$40 \cdot 2^{1} + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.2.5

Evaluate the exponent.

$40 \cdot 2 + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.3

Multiply $40$ by $2$ .

$80 + 10 \sqrt{2} \cdot - 2 + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.4

Multiply $- 2$ by $10$ .

$80 - 20 \sqrt{2} + 5 (4 \sqrt{2}) + 5 \cdot - 2$

Step 3.1.5

Multiply $4$ by $5$ .

$80 - 20 \sqrt{2} + 20 \sqrt{2} + 5 \cdot - 2$

Step 3.1.6

Multiply $5$ by $- 2$ .

$80 - 20 \sqrt{2} + 20 \sqrt{2} - 10$

Step 3.2

Subtract $10$ from $80$ .

$70 - 20 \sqrt{2} + 20 \sqrt{2}$

Step 3.3

Add $- 20 \sqrt{2}$ and $20 \sqrt{2}$ .

$70 + 0$

Step 3.4

Add $70$ and $0$ .

$70$