Simplify |(1+i)^n|

$| {(1 + i)}^{n} |$

Step 1

Add $0$ .

$| {(1 + i)}^{n} + 0 |$

Step 2

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

$\sqrt{{({(1 + i)}^{n})}^{2} + 0^{2}}$

Step 3

Multiply the exponents in ${({(1 + i)}^{n})}^{2}$ .

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

$\sqrt{{(1 + i)}^{n \cdot 2} + 0^{2}}$

Move $2$ to the left of $n$ .

$\sqrt{{(1 + i)}^{2 n} + 0^{2}}$

Step 4

Raising $0$ to any positive power yields $0$ .

$\sqrt{{(1 + i)}^{2 n} + 0}$

Step 5

Add ${(1 + i)}^{2 n}$ and $0$ .

$\sqrt{{(1 + i)}^{2 n}}$

Step 6

Rewrite ${(1 + i)}^{2 n}$ as ${({(1 + i)}^{n})}^{2}$ .

$\sqrt{{({(1 + i)}^{n})}^{2}}$

Step 7

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

${(1 + i)}^{n}$