Trigonometry Examples

$v ({(410 + 140 \sqrt{2})}^{2} + {(140 \sqrt{2})}^{2})$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(410 + 140 \sqrt{2})}^{2}$ as $(410 + 140 \sqrt{2}) (410 + 140 \sqrt{2})$ .

$v ((410 + 140 \sqrt{2}) (410 + 140 \sqrt{2}) + {(140 \sqrt{2})}^{2})$

Step 1.2

Expand $(410 + 140 \sqrt{2}) (410 + 140 \sqrt{2})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$v (410 (410 + 140 \sqrt{2}) + 140 \sqrt{2} (410 + 140 \sqrt{2}) + {(140 \sqrt{2})}^{2})$

Step 1.2.2

Apply the distributive property.

$v (410 \cdot 410 + 410 (140 \sqrt{2}) + 140 \sqrt{2} (410 + 140 \sqrt{2}) + {(140 \sqrt{2})}^{2})$

Step 1.2.3

Apply the distributive property.

$v (410 \cdot 410 + 410 (140 \sqrt{2}) + 140 \sqrt{2} \cdot 410 + 140 \sqrt{2} (140 \sqrt{2}) + {(140 \sqrt{2})}^{2})$

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 $410$ by $410$ .

$v (168100 + 410 (140 \sqrt{2}) + 140 \sqrt{2} \cdot 410 + 140 \sqrt{2} (140 \sqrt{2}) + {(140 \sqrt{2})}^{2})$

Step 1.3.1.2

Multiply $140$ by $410$ .

$v (168100 + 57400 \sqrt{2} + 140 \sqrt{2} \cdot 410 + 140 \sqrt{2} (140 \sqrt{2}) + {(140 \sqrt{2})}^{2})$

Step 1.3.1.3

Multiply $410$ by $140$ .

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 140 \sqrt{2} (140 \sqrt{2}) + {(140 \sqrt{2})}^{2})$

Step 1.3.1.4

Multiply $140 \sqrt{2} (140 \sqrt{2})$ .

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Step 1.3.1.4.1

Multiply $140$ by $140$ .

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 \sqrt{2} \sqrt{2} + {(140 \sqrt{2})}^{2})$

Step 1.3.1.4.2

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

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 ({\sqrt{2}}^{1} \sqrt{2}) + {(140 \sqrt{2})}^{2})$

Step 1.3.1.4.3

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

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + {(140 \sqrt{2})}^{2})$

Step 1.3.1.4.4

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

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 {\sqrt{2}}^{1 + 1} + {(140 \sqrt{2})}^{2})$

Step 1.3.1.4.5

Add $1$ and $1$ .

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 {\sqrt{2}}^{2} + {(140 \sqrt{2})}^{2})$

Step 1.3.1.5

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

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Step 1.3.1.5.1

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

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 {(2^{\frac{1}{2}})}^{2} + {(140 \sqrt{2})}^{2})$

Step 1.3.1.5.2

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

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 \cdot 2^{\frac{1}{2} \cdot 2} + {(140 \sqrt{2})}^{2})$

Step 1.3.1.5.3

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

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 \cdot 2^{\frac{2}{2}} + {(140 \sqrt{2})}^{2})$

Step 1.3.1.5.4

Cancel the common factor of $2$ .

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Step 1.3.1.5.4.1

Cancel the common factor.

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 \cdot 2^{\frac{2}{2}} + {(140 \sqrt{2})}^{2})$

Step 1.3.1.5.4.2

Rewrite the expression.

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 \cdot 2^{1} + {(140 \sqrt{2})}^{2})$

Step 1.3.1.5.5

Evaluate the exponent.

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 19600 \cdot 2 + {(140 \sqrt{2})}^{2})$

Step 1.3.1.6

Multiply $19600$ by $2$ .

$v (168100 + 57400 \sqrt{2} + 57400 \sqrt{2} + 39200 + {(140 \sqrt{2})}^{2})$

Step 1.3.2

Add $168100$ and $39200$ .

$v (207300 + 57400 \sqrt{2} + 57400 \sqrt{2} + {(140 \sqrt{2})}^{2})$

Step 1.3.3

Add $57400 \sqrt{2}$ and $57400 \sqrt{2}$ .

$v (207300 + 114800 \sqrt{2} + {(140 \sqrt{2})}^{2})$

Step 1.4

Apply the product rule to $140 \sqrt{2}$ .

$v (207300 + 114800 \sqrt{2} + 140^{2} {\sqrt{2}}^{2})$

Step 1.5

Raise $140$ to the power of $2$ .

$v (207300 + 114800 \sqrt{2} + 19600 {\sqrt{2}}^{2})$

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 $2^{\frac{1}{2}}$ .

$v (207300 + 114800 \sqrt{2} + 19600 {(2^{\frac{1}{2}})}^{2})$

Step 1.6.2

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

$v (207300 + 114800 \sqrt{2} + 19600 \cdot 2^{\frac{1}{2} \cdot 2})$

Step 1.6.3

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

$v (207300 + 114800 \sqrt{2} + 19600 \cdot 2^{\frac{2}{2}})$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$v (207300 + 114800 \sqrt{2} + 19600 \cdot 2^{\frac{2}{2}})$

Step 1.6.4.2

Rewrite the expression.

$v (207300 + 114800 \sqrt{2} + 19600 \cdot 2^{1})$

Step 1.6.5

Evaluate the exponent.

$v (207300 + 114800 \sqrt{2} + 19600 \cdot 2)$

Step 1.7

Multiply $19600$ by $2$ .

$v (207300 + 114800 \sqrt{2} + 39200)$

Step 2

Simplify by multiplying through.

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Step 2.1

Add $207300$ and $39200$ .

$v (246500 + 114800 \sqrt{2})$

Step 2.2

Apply the distributive property.

$v \cdot 246500 + v (114800 \sqrt{2})$

Step 2.3

Move $246500$ to the left of $v$ .

$246500 \cdot v + v (114800 \sqrt{2})$

Step 3

Move $114800$ to the left of $v$ .

$246500 v + 114800 v \sqrt{2}$