Trigonometry Examples

$\sqrt{{(45 \sqrt{2})}^{2} + {(650 + 45 \sqrt{2})}^{2}}$

Step 1

Simplify by cancelling exponent with radical.

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Step 1.1

Apply the product rule to $45 \sqrt{2}$ .

$\sqrt{45^{2} {\sqrt{2}}^{2} + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.2

Raise $45$ to the power of $2$ .

$\sqrt{2025 {\sqrt{2}}^{2} + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.3

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

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Step 1.3.1

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

$\sqrt{2025 {(2^{\frac{1}{2}})}^{2} + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.3.2

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

$\sqrt{2025 \cdot 2^{\frac{1}{2} \cdot 2} + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.3.3

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

$\sqrt{2025 \cdot 2^{\frac{2}{2}} + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

$\sqrt{2025 \cdot 2^{\frac{2}{2}} + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.3.4.2

Rewrite the expression.

$\sqrt{2025 \cdot 2^{1} + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.3.5

Evaluate the exponent.

$\sqrt{2025 \cdot 2 + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.4

Simplify the expression.

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Step 1.4.1

Multiply $2025$ by $2$ .

$\sqrt{4050 + {(650 + 45 \sqrt{2})}^{2}}$

Step 1.4.2

Rewrite ${(650 + 45 \sqrt{2})}^{2}$ as $(650 + 45 \sqrt{2}) (650 + 45 \sqrt{2})$ .

$\sqrt{4050 + (650 + 45 \sqrt{2}) (650 + 45 \sqrt{2})}$

Step 2

Expand $(650 + 45 \sqrt{2}) (650 + 45 \sqrt{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{4050 + 650 (650 + 45 \sqrt{2}) + 45 \sqrt{2} (650 + 45 \sqrt{2})}$

Step 2.2

Apply the distributive property.

$\sqrt{4050 + 650 \cdot 650 + 650 (45 \sqrt{2}) + 45 \sqrt{2} (650 + 45 \sqrt{2})}$

Step 2.3

Apply the distributive property.

$\sqrt{4050 + 650 \cdot 650 + 650 (45 \sqrt{2}) + 45 \sqrt{2} \cdot 650 + 45 \sqrt{2} (45 \sqrt{2})}$

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 $650$ by $650$ .

$\sqrt{4050 + 422500 + 650 (45 \sqrt{2}) + 45 \sqrt{2} \cdot 650 + 45 \sqrt{2} (45 \sqrt{2})}$

Step 3.1.2

Multiply $45$ by $650$ .

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 45 \sqrt{2} \cdot 650 + 45 \sqrt{2} (45 \sqrt{2})}$

Step 3.1.3

Multiply $650$ by $45$ .

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 45 \sqrt{2} (45 \sqrt{2})}$

Step 3.1.4

Multiply $45 \sqrt{2} (45 \sqrt{2})$ .

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Step 3.1.4.1

Multiply $45$ by $45$ .

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 \sqrt{2} \sqrt{2}}$

Step 3.1.4.2

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

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 3.1.4.3

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

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 3.1.4.4

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

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 {\sqrt{2}}^{1 + 1}}$

Step 3.1.4.5

Add $1$ and $1$ .

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 {\sqrt{2}}^{2}}$

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

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 {(2^{\frac{1}{2}})}^{2}}$

Step 3.1.5.2

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

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.1.5.3

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

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 \cdot 2^{\frac{2}{2}}}$

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.

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 \cdot 2^{\frac{2}{2}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 \cdot 2^{1}}$

Step 3.1.5.5

Evaluate the exponent.

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 2025 \cdot 2}$

Step 3.1.6

Multiply $2025$ by $2$ .

$\sqrt{4050 + 422500 + 29250 \sqrt{2} + 29250 \sqrt{2} + 4050}$

Step 3.2

Add $422500$ and $4050$ .

$\sqrt{4050 + 426550 + 29250 \sqrt{2} + 29250 \sqrt{2}}$

Step 3.3

Add $29250 \sqrt{2}$ and $29250 \sqrt{2}$ .

$\sqrt{4050 + 426550 + 58500 \sqrt{2}}$

Step 4

Add $4050$ and $426550$ .

$\sqrt{430600 + 58500 \sqrt{2}}$

Step 5

Rewrite $430600 + 58500 \sqrt{2}$ as $10^{2} (4306 + 585 \sqrt{2})$ .

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Step 5.1

Factor $100$ out of $430600$ .

$\sqrt{100 (4306) + 58500 \sqrt{2}}$

Step 5.2

Factor $100$ out of $58500 \sqrt{2}$ .

$\sqrt{100 (4306) + 100 (585 \sqrt{2})}$

Step 5.3

Factor $100$ out of $100 (4306) + 100 (585 \sqrt{2})$ .

$\sqrt{100 (4306 + 585 \sqrt{2})}$

Step 5.4

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2} (4306 + 585 \sqrt{2})}$

Step 6

Pull terms out from under the radical.

$10 \sqrt{4306 + 585 \sqrt{2}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt{4306 + 585 \sqrt{2}}$

Decimal Form:

$716.47155798 \dots$