Problem 81 Simplify. $$i^{35}$$... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 81

Simplify. $$i^{35}$$

Short Answer

Expert verified

$i^{35} = -i$

Step by step solution

Understand the powers of i

The imaginary unit, denoted as i, has cyclical powers. The sequence repeats every four powers: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. Beyond this, the pattern repeats.

Determine the equivalent exponent within one cycle

To find an equivalent exponent for $i^{35}$ within one cycle of 4, compute the remainder when 35 is divided by 4: $35 \, \text{mod} \, 4 = 3$. Therefore, $i^{35} = i^{3}$.

Simplify using the cyclical pattern

Given the cyclical nature identified earlier, $i^3 = -i$. Thus, $i^{35} = -i$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Numbers

Imaginary numbers help us simplify equations that don't have solutions in the real numbers domain.
The imaginary unit is denoted by the symbol 'i', and it's defined by the equation: $i^2 = -1$. This means if you square the imaginary unit, you get -1.
Combining imaginary and real numbers, we get complex numbers, which take the form a + bi, where 'a' and 'b' are real numbers.
This is useful in many areas of math and engineering.

Cyclical Patterns

The powers of the imaginary unit follow a cyclical pattern of every four steps.
Here is how it works:
\i^1 = i\, \i^2 = -1\, \i^3 = -i\, \i^4 = 1\, and then it starts over. This cycle makes it much easier to work with large exponents of i.
Instead of calculating each power from scratch, we use the cyclical pattern to simplify our work.
For instance, $i^{35}$ can be simplified by finding where it falls in the pattern. This is where the modulus operation comes in.

Modulus Operation

The modulus operation helps find the remainder when one number is divided by another.
It's represented as 'mod'. This operation is very useful when dealing with cyclical patterns.
For example, to simplify $i^{35}$, you can find the remainder of 35 divided by 4:

35 mod 4 = 3

This tells us that $i^{35}$ is the same as $i^3$.
From the cyclical pattern, we know $i^3 = -i\.$. So, $i^{35} = -i\.$.
Using the modulus operation makes calculations with imaginary numbers much easier.

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91影视

Simplify. $$i^{35}$$

Short Answer

Step by step solution

Understand the powers of i

Determine the equivalent exponent within one cycle

Simplify using the cyclical pattern

Key Concepts

Imaginary Numbers

Cyclical Patterns

Modulus Operation

One App. One Place for Learning.

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