Simplify: i^{-10} How exactly do I solve this?

Question:

Simplify: {eq}i^{-10} {/eq} How exactly do I solve this?

Imaginary number:

Imaginary numbers are denoted by the letter {eq}i {/eq} and represents the concept of the square root of negative one such that {eq}i = \sqrt{-1} {/eq}. Any term with an imaginary number is called a complex number.

Answer and Explanation:

Simplify: {eq}i^{-10} {/eq} How exactly do I solve this?

To get the simplified form of the expression below, express it as a radical and use the power rule of exponents:

{eq}i^{-10} = (\sqrt {-1})^{10} {/eq}

{eq}(\sqrt {-1})^{10} = ((-1)^{\frac12})^{10} {/eq}

{eq}((-1)^{\frac12})^{10} = {(-1)}^{\frac12 \cdot 10} {/eq}

{eq}(-1)^{\frac12 \cdot 10} = (-1)^{\frac {10} 2 } {/eq}

{eq}(-1)^{\frac {10} 2 } = (-1)^{5 } {/eq}

{eq}(-1)^{5 } = -1 {/eq}

The expression {eq}i^{-10} {/eq} when simplified is {eq}-1 {/eq}.


Learn more about this topic:

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What is an Imaginary Number?

from Math 101: College Algebra

Chapter 5 / Lesson 1
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