Trigonometry Examples

$\sqrt{{(50 \sqrt{2})}^{2} + {(400 + 50 \sqrt{2})}^{2}}$

Step 1

Simplify by cancelling exponent with radical.

Tap for more steps...

Step 1.1

Apply the product rule to $50 \sqrt{2}$ .

$\sqrt{50^{2} {\sqrt{2}}^{2} + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.2

Raise $50$ to the power of $2$ .

$\sqrt{2500 {\sqrt{2}}^{2} + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$\sqrt{2500 {(2^{\frac{1}{2}})}^{2} + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.3.2

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

$\sqrt{2500 \cdot 2^{\frac{1}{2} \cdot 2} + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.3.3

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

$\sqrt{2500 \cdot 2^{\frac{2}{2}} + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.4.1

Cancel the common factor.

$\sqrt{2500 \cdot 2^{\frac{2}{2}} + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.3.4.2

Rewrite the expression.

$\sqrt{2500 \cdot 2^{1} + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.3.5

Evaluate the exponent.

$\sqrt{2500 \cdot 2 + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.4

Simplify the expression.

Tap for more steps...

Step 1.4.1

Multiply $2500$ by $2$ .

$\sqrt{5000 + {(400 + 50 \sqrt{2})}^{2}}$

Step 1.4.2

Rewrite ${(400 + 50 \sqrt{2})}^{2}$ as $(400 + 50 \sqrt{2}) (400 + 50 \sqrt{2})$ .

$\sqrt{5000 + (400 + 50 \sqrt{2}) (400 + 50 \sqrt{2})}$

Step 2

Expand $(400 + 50 \sqrt{2}) (400 + 50 \sqrt{2})$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

$\sqrt{5000 + 400 (400 + 50 \sqrt{2}) + 50 \sqrt{2} (400 + 50 \sqrt{2})}$

Step 2.2

Apply the distributive property.

$\sqrt{5000 + 400 \cdot 400 + 400 (50 \sqrt{2}) + 50 \sqrt{2} (400 + 50 \sqrt{2})}$

Step 2.3

Apply the distributive property.

$\sqrt{5000 + 400 \cdot 400 + 400 (50 \sqrt{2}) + 50 \sqrt{2} \cdot 400 + 50 \sqrt{2} (50 \sqrt{2})}$

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 $400$ by $400$ .

$\sqrt{5000 + 160000 + 400 (50 \sqrt{2}) + 50 \sqrt{2} \cdot 400 + 50 \sqrt{2} (50 \sqrt{2})}$

Step 3.1.2

Multiply $50$ by $400$ .

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 50 \sqrt{2} \cdot 400 + 50 \sqrt{2} (50 \sqrt{2})}$

Step 3.1.3

Multiply $400$ by $50$ .

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 50 \sqrt{2} (50 \sqrt{2})}$

Step 3.1.4

Multiply $50 \sqrt{2} (50 \sqrt{2})$ .

Tap for more steps...

Step 3.1.4.1

Multiply $50$ by $50$ .

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 \sqrt{2} \sqrt{2}}$

Step 3.1.4.2

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

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 3.1.4.3

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

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 3.1.4.4

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

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 {\sqrt{2}}^{1 + 1}}$

Step 3.1.4.5

Add $1$ and $1$ .

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 {\sqrt{2}}^{2}}$

Step 3.1.5

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

Tap for more steps...

Step 3.1.5.1

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

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 {(2^{\frac{1}{2}})}^{2}}$

Step 3.1.5.2

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

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.1.5.3

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

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 \cdot 2^{\frac{2}{2}}}$

Step 3.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.5.4.1

Cancel the common factor.

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 \cdot 2^{\frac{2}{2}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 \cdot 2^{1}}$

Step 3.1.5.5

Evaluate the exponent.

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 2500 \cdot 2}$

Step 3.1.6

Multiply $2500$ by $2$ .

$\sqrt{5000 + 160000 + 20000 \sqrt{2} + 20000 \sqrt{2} + 5000}$

Step 3.2

Add $160000$ and $5000$ .

$\sqrt{5000 + 165000 + 20000 \sqrt{2} + 20000 \sqrt{2}}$

Step 3.3

Add $20000 \sqrt{2}$ and $20000 \sqrt{2}$ .

$\sqrt{5000 + 165000 + 40000 \sqrt{2}}$

Step 4

Add $5000$ and $165000$ .

$\sqrt{170000 + 40000 \sqrt{2}}$

Step 5

Rewrite $170000 + 40000 \sqrt{2}$ as $100^{2} (17 + 4 \sqrt{2})$ .

Tap for more steps...

Step 5.1

Factor $10000$ out of $170000$ .

$\sqrt{10000 (17) + 40000 \sqrt{2}}$

Step 5.2

Factor $10000$ out of $40000 \sqrt{2}$ .

$\sqrt{10000 (17) + 10000 (4 \sqrt{2})}$

Step 5.3

Factor $10000$ out of $10000 (17) + 10000 (4 \sqrt{2})$ .

$\sqrt{10000 (17 + 4 \sqrt{2})}$

Step 5.4

Rewrite $10000$ as $100^{2}$ .

$\sqrt{100^{2} (17 + 4 \sqrt{2})}$

Step 6

Pull terms out from under the radical.

$100 \sqrt{17 + 4 \sqrt{2}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$100 \sqrt{17 + 4 \sqrt{2}}$

Decimal Form:

$475.99216642 \dots$