Trigonometry Examples

$v ({(500 + 60 \sqrt{2})}^{2})$

Step 1

Rewrite ${(500 + 60 \sqrt{2})}^{2}$ as $(500 + 60 \sqrt{2}) (500 + 60 \sqrt{2})$ .

$v ((500 + 60 \sqrt{2}) (500 + 60 \sqrt{2}))$

Step 2

Expand $(500 + 60 \sqrt{2}) (500 + 60 \sqrt{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$v (500 (500 + 60 \sqrt{2}) + 60 \sqrt{2} (500 + 60 \sqrt{2}))$

Step 2.2

Apply the distributive property.

$v (500 \cdot 500 + 500 (60 \sqrt{2}) + 60 \sqrt{2} (500 + 60 \sqrt{2}))$

Step 2.3

Apply the distributive property.

$v (500 \cdot 500 + 500 (60 \sqrt{2}) + 60 \sqrt{2} \cdot 500 + 60 \sqrt{2} (60 \sqrt{2}))$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $500$ by $500$ .

$v (250000 + 500 (60 \sqrt{2}) + 60 \sqrt{2} \cdot 500 + 60 \sqrt{2} (60 \sqrt{2}))$

Step 3.1.2

Multiply $60$ by $500$ .

$v (250000 + 30000 \sqrt{2} + 60 \sqrt{2} \cdot 500 + 60 \sqrt{2} (60 \sqrt{2}))$

Step 3.1.3

Multiply $500$ by $60$ .

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 60 \sqrt{2} (60 \sqrt{2}))$

Step 3.1.4

Multiply $60 \sqrt{2} (60 \sqrt{2})$ .

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Step 3.1.4.1

Multiply $60$ by $60$ .

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 \sqrt{2} \sqrt{2})$

Step 3.1.4.2

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

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 ({\sqrt{2}}^{1} \sqrt{2}))$

Step 3.1.4.3

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

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}))$

Step 3.1.4.4

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

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 {\sqrt{2}}^{1 + 1})$

Step 3.1.4.5

Add $1$ and $1$ .

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 {\sqrt{2}}^{2})$

Step 3.1.5

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

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Step 3.1.5.1

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

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 {(2^{\frac{1}{2}})}^{2})$

Step 3.1.5.2

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

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 \cdot 2^{\frac{1}{2} \cdot 2})$

Step 3.1.5.3

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

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 \cdot 2^{\frac{2}{2}})$

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 \cdot 2^{\frac{2}{2}})$

Step 3.1.5.4.2

Rewrite the expression.

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 \cdot 2^{1})$

Step 3.1.5.5

Evaluate the exponent.

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 3600 \cdot 2)$

Step 3.1.6

Multiply $3600$ by $2$ .

$v (250000 + 30000 \sqrt{2} + 30000 \sqrt{2} + 7200)$

Step 3.2

Add $250000$ and $7200$ .

$v (257200 + 30000 \sqrt{2} + 30000 \sqrt{2})$

Step 3.3

Add $30000 \sqrt{2}$ and $30000 \sqrt{2}$ .

$v (257200 + 60000 \sqrt{2})$

Step 4

Simplify by multiplying through.

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Step 4.1

Apply the distributive property.

$v \cdot 257200 + v (60000 \sqrt{2})$

Step 4.2

Move $257200$ to the left of $v$ .

$257200 \cdot v + v (60000 \sqrt{2})$

Step 5

Move $60000$ to the left of $v$ .

$257200 v + 60000 v \sqrt{2}$