Trigonometry Examples

$v ({(- 250 \sqrt{2} + 100)}^{2} + {(- 250 \sqrt{2})}^{2})$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(- 250 \sqrt{2} + 100)}^{2}$ as $(- 250 \sqrt{2} + 100) (- 250 \sqrt{2} + 100)$ .

$v ((- 250 \sqrt{2} + 100) (- 250 \sqrt{2} + 100) + {(- 250 \sqrt{2})}^{2})$

Step 1.2

Expand $(- 250 \sqrt{2} + 100) (- 250 \sqrt{2} + 100)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$v (- 250 \sqrt{2} (- 250 \sqrt{2} + 100) + 100 (- 250 \sqrt{2} + 100) + {(- 250 \sqrt{2})}^{2})$

Step 1.2.2

Apply the distributive property.

$v (- 250 \sqrt{2} (- 250 \sqrt{2}) - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2} + 100) + {(- 250 \sqrt{2})}^{2})$

Step 1.2.3

Apply the distributive property.

$v (- 250 \sqrt{2} (- 250 \sqrt{2}) - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $- 250 \sqrt{2} (- 250 \sqrt{2})$ .

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Step 1.3.1.1.1

Multiply $- 250$ by $- 250$ .

$v (62500 \sqrt{2} \sqrt{2} - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.1.2

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

$v (62500 ({\sqrt{2}}^{1} \sqrt{2}) - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.1.3

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

$v (62500 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.1.4

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

$v (62500 {\sqrt{2}}^{1 + 1} - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.1.5

Add $1$ and $1$ .

$v (62500 {\sqrt{2}}^{2} - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.2

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

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Step 1.3.1.2.1

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

$v (62500 {(2^{\frac{1}{2}})}^{2} - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.2.2

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

$v (62500 \cdot 2^{\frac{1}{2} \cdot 2} - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.2.3

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

$v (62500 \cdot 2^{\frac{2}{2}} - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.2.4

Cancel the common factor of $2$ .

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Step 1.3.1.2.4.1

Cancel the common factor.

$v (62500 \cdot 2^{\frac{2}{2}} - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.2.4.2

Rewrite the expression.

$v (62500 \cdot 2^{1} - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.2.5

Evaluate the exponent.

$v (62500 \cdot 2 - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.3

Multiply $62500$ by $2$ .

$v (125000 - 250 \sqrt{2} \cdot 100 + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.4

Multiply $100$ by $- 250$ .

$v (125000 - 25000 \sqrt{2} + 100 (- 250 \sqrt{2}) + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.5

Multiply $- 250$ by $100$ .

$v (125000 - 25000 \sqrt{2} - 25000 \sqrt{2} + 100 \cdot 100 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.1.6

Multiply $100$ by $100$ .

$v (125000 - 25000 \sqrt{2} - 25000 \sqrt{2} + 10000 + {(- 250 \sqrt{2})}^{2})$

Step 1.3.2

Add $125000$ and $10000$ .

$v (135000 - 25000 \sqrt{2} - 25000 \sqrt{2} + {(- 250 \sqrt{2})}^{2})$

Step 1.3.3

Subtract $25000 \sqrt{2}$ from $- 25000 \sqrt{2}$ .

$v (135000 - 50000 \sqrt{2} + {(- 250 \sqrt{2})}^{2})$

Step 1.4

Apply the product rule to $- 250 \sqrt{2}$ .

$v (135000 - 50000 \sqrt{2} + {(- 250)}^{2} {\sqrt{2}}^{2})$

Step 1.5

Raise $- 250$ to the power of $2$ .

$v (135000 - 50000 \sqrt{2} + 62500 {\sqrt{2}}^{2})$

Step 1.6

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

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Step 1.6.1

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

$v (135000 - 50000 \sqrt{2} + 62500 {(2^{\frac{1}{2}})}^{2})$

Step 1.6.2

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

$v (135000 - 50000 \sqrt{2} + 62500 \cdot 2^{\frac{1}{2} \cdot 2})$

Step 1.6.3

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

$v (135000 - 50000 \sqrt{2} + 62500 \cdot 2^{\frac{2}{2}})$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$v (135000 - 50000 \sqrt{2} + 62500 \cdot 2^{\frac{2}{2}})$

Step 1.6.4.2

Rewrite the expression.

$v (135000 - 50000 \sqrt{2} + 62500 \cdot 2^{1})$

Step 1.6.5

Evaluate the exponent.

$v (135000 - 50000 \sqrt{2} + 62500 \cdot 2)$

Step 1.7

Multiply $62500$ by $2$ .

$v (135000 - 50000 \sqrt{2} + 125000)$

Step 2

Simplify by multiplying through.

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Step 2.1

Add $135000$ and $125000$ .

$v (260000 - 50000 \sqrt{2})$

Step 2.2

Apply the distributive property.

$v \cdot 260000 + v (- 50000 \sqrt{2})$

Step 2.3

Simplify the expression.

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Step 2.3.1

Move $260000$ to the left of $v$ .

$260000 \cdot v + v (- 50000 \sqrt{2})$

Step 2.3.2

Reorder factors in $260000 v + v (- 50000 \sqrt{2})$ .

$260000 v - 50000 v \sqrt{2}$