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.

Skip topic navigation

1. Review of Real Numbers2h 39m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
40m

2. Linear Equations and Inequalities3h 38m

The Addition and Subtraction Properties of Equality
41m

The Multiplication and Division Properties of Equality
30m

Solving Linear Equations
1h 14m

Linear Inequalities in One Variable
1h 11m

3. Solving Word Problems2h 43m

Introduction to Problem Solving
37m

Formulas
21m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphing Linear Equations in Two Variables3h 17m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

5. Systems of Linear Equations1h 43m

Solving Systems of Linear Equations by Graphing
41m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

6. Exponents and Polynomials3h 25m

The Product Rule
10m

The Quotient Rule
13m

Intro to the Power Rules
18m

The Power of a Quotient Rule
18m

Negative Exponents
28m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
33m

Special Products
34m

7. Factoring2h 42m

Factoring by Greatest Common Factor & Grouping
37m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
33m

Quadratic Equations & Applications
41m

8. Rational Expressions and Equations3h 13m

Simplifying Rational Expressions
39m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

9. Inequalities and Absolute Value2h 52m

Set Operations and Compound Inequalities
42m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

Linear Inequalities in Two Variables
28m

Systems of Linear Inequalities
23m

10. Relations and Functions2h 9m

Introduction to Relations and Functions
57m

Function Notation
15m

Graphing Common Functions
5m

Composition of Functions
24m

Direct & Inverse Variation
27m

11. Roots, Radicals, and Complex Numbers3h 57m

Introduction to Radical Expressions
47m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

12. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

13. Inverse, Exponential, & Logarithmic Functions2h 30m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
54m

14. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

15. Sequences, Series, and the Binomial Theorem1h 48m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
26m

Factorials
13m

11. Roots, Radicals, and Complex Numbers

Multiplying and Dividing Complex Numbers

Beginning & Intermediate Algebra

11. Roots, Radicals, and Complex Numbers

Multiplying and Dividing Complex Numbers

Previous problemNext problem

Struggling with Beginning & Intermediate Algebra?

Join thousands of students who trust us to help them ace their exams!

Multiple Choice

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

$64 i cubed$

$64 i$

$1 over 64 i$

$1 over 64 i squared$

0 댓글

Previous problemNext problem

검증된 단계별 안내

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.

5:33m

Watch next

Master Complex Conjugates with a bite sized video explanation from Patrick Ford

Struggling with Beginning & Intermediate Algebra?

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

Watch next

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