Trigonometry Examples

$\sqrt{{(3121)}^{2} + {(1500 \sqrt{2})}^{2}}$

Step 1

Simplify the expression.

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Step 1.1

Raise $3121$ to the power of $2$ .

$\sqrt{9740641 + {(1500 \sqrt{2})}^{2}}$

Step 1.2

Apply the product rule to $1500 \sqrt{2}$ .

$\sqrt{9740641 + 1500^{2} {\sqrt{2}}^{2}}$

Step 1.3

Raise $1500$ to the power of $2$ .

$\sqrt{9740641 + 2250000 {\sqrt{2}}^{2}}$

Step 2

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

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Step 2.1

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

$\sqrt{9740641 + 2250000 {(2^{\frac{1}{2}})}^{2}}$

Step 2.2

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

$\sqrt{9740641 + 2250000 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.3

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

$\sqrt{9740641 + 2250000 \cdot 2^{\frac{2}{2}}}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$\sqrt{9740641 + 2250000 \cdot 2^{\frac{2}{2}}}$

Step 2.4.2

Rewrite the expression.

$\sqrt{9740641 + 2250000 \cdot 2^{1}}$

Step 2.5

Evaluate the exponent.

$\sqrt{9740641 + 2250000 \cdot 2}$

Step 3

Simplify the expression.

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Step 3.1

Multiply $2250000$ by $2$ .

$\sqrt{9740641 + 4500000}$

Step 3.2

Add $9740641$ and $4500000$ .

$\sqrt{14240641}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{14240641}$

Decimal Form:

$3773.67738419 \dots$