Evaluate the following powers of i i . ( 4 i ) − 3 \left(4i\right)^{-3}

we're asked to evaluate the following powers of I. We're given here four I raised to the power of negative three. Because this is a negative exponent, I'm going to go ahead and rewrite this as the denominator of a fraction. So this becomes one over four I cubed with that exponent no longer negative since I put it on the bottom of my fraction. Now I can go ahead and distribute that power to both four and I. So that makes this one over four cubed times I cubed. Now four cubed is 64. So this is one over 64 times I cubed. Well, let's think about what I cubed is. It's I times I times I. I times I is negative one times I will just give me negative one I or negative I. So this is 64 times negative I. I'm gonna stick that negative on the front here. So this is negative 64 I. Now we could stop here, but remember that we don't want these imaginary values in the bottom of our fraction. We don't want it in the denominator. So we wanna rationalize our denominator here. And we can do this by multiplying by I over I. Because once I have an I squared, I'll no longer have that I in the denominator. I'll just have a negative one. So let's see what happens when we do this multiplication here. I times I in the bottom will give me that negative one. In the top, I just have one times I, which is I. So this is negative 64 times I squared, which remember is just negative one. So I have negative 64 times negative one, which will give me positive 64. So I have I over 64. And typically, when working with complex values or imaginary values, we really just want to write it as a coefficient to I rather than division. So I would rewrite this as one over 64 I. And here is my solution, having simplified it as well. Let us know if you have any questions, and I'll see you soon.

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Multiplying and Dividing Complex Numbers

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Multiple Choice

Evaluate the following powers of $i$ .
${(4 i)}^{- 3}$

$64 i cubed$

$64 i$

$1 over 64 i$

$1 over 64 i squared$

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Verified step by step guidance

Rewrite the expression with the negative exponent as a reciprocal: $\left(4i\right)^{-3} = \frac{1}{\left(4i\right)^3}$.

Calculate the cube of the product inside the denominator: $\left(4i\right)^3 = 4^3 \cdot i^3$.

Evaluate the numerical part: \$4^3 = 64$, so the expression becomes $\frac{1}{64 \cdot i^3}$.

Recall the powers of $i$: $i^1 = i$, $i^2 = -1$, $i^3 = i^2 \cdot i = -1 \cdot i = -i$.

Substitute $i^3$ with $-i$ in the denominator and simplify the fraction accordingly.

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Evaluate the following powers of ii.(4i)−3\(\left\)(4i\(\right\))^{-3}

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Evaluate the following powers of $i$ .
${(4 i)}^{- 3}$