Algebra Examples

$| \sqrt{3} + i |$

Step 1

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{{\sqrt{3}}^{2} + 1^{2}}$

Step 2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{{(3^{\frac{1}{2}})}^{2} + 1^{2}}$

Step 2.2

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

$\sqrt{3^{\frac{1}{2} \cdot 2} + 1^{2}}$

Step 2.3

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

$\sqrt{3^{\frac{2}{2}} + 1^{2}}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$\sqrt{3^{\frac{2}{2}} + 1^{2}}$

Step 2.4.2

Rewrite the expression.

$\sqrt{3^{1} + 1^{2}}$

Step 2.5

Evaluate the exponent.

$\sqrt{3 + 1^{2}}$

Step 3

One to any power is one.

$\sqrt{3 + 1}$

Step 4

Add $3$ and $1$ .

$\sqrt{4}$

Step 5

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2}}$

Step 6

Pull terms out from under the radical, assuming positive real numbers.

$2$