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Chapter 3: Problem 80

Simplify. $$i^{7}$$

Short Answer

Expert verified
i^{7} = -i

Step by step solution

01

- Understand the Powers of i

Recall that the imaginary unit, denoted as i, has a cyclic pattern in its powers. Specifically: i^1 = ii^2 = -1i^3 = -ii^4 = 1i^5 = ii^6 = -1i^7 = -ii^8 = 1i^n repeats every 4th power, so we can use this cycle to simplify higher powers of i.
02

- Find the Remainder When Dividing by 4

Given the cyclic pattern, divide the exponent by 4 and find the remainder. This will tell which part of the cycle corresponds to the given power.For i^7, divide 7 by 4:7 梅 4 = 1 with a remainder of 3.So, i^7 corresponds to i^3.
03

- Use the Cycle to Simplify

From the cycle of powers of i, we know that:i^3 = -iTherefore, i^7 = -i.

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Key Concepts

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

Imaginary Unit
The concept of the imaginary unit, denoted as \(i\), is fundamental in understanding complex numbers.

Simply put, the imaginary unit is defined as \(i = \sqrt{-1}\).
This means that when you square the imaginary unit, you get -1:

\(i^2 = -1\).

The power of the imaginary unit grows increasingly important in solving equations and simplifying expressions that involve complex numbers.
Cyclic Pattern
One of the most useful aspects of dealing with the imaginary unit is recognizing its cyclic pattern in its powers.
The pattern repeats every four exponents:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)

After \(i^4\), the pattern starts over again, making calculations easier. Understanding this cycle can help you simplify higher powers of \(i\) quickly.
Exponent Division
Once we understand the cyclic pattern of the powers of \(i\), we can simplify any power of \(i\) by dividing the exponent by 4 and finding the remainder.

For example, to simplify \(i^7\), you divide 7 by 4, giving a quotient of 1 and a remainder of 3:

\(7 \div 4 = 1\) remainder 3.

This tells us that \(i^7\) is equivalent to \(i^3\). From the cyclic pattern, we know \(i^3 = -i\). So, \(i^7 = -i\).

This approach can be used for any exponent, making the process of simplifying powers of \(i\) straightforward and efficient.

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Most popular questions from this chapter

In business, profit is the difference between revenue and cost; that is, $$ \begin{array}{l} \text { Total profit }=\text { Total revenue }-\text { Total cost, } \\ P(x)=R(x)-C(x) \end{array} $$ where \(x\) is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for each of the following. $$R(x)=5 x, C(x)=0.001 x^{2}+1.2 x+60$$

Determine whether the statement is true or false. The conjugate of a product is the product of the conjugates of the individual complex numbers.

Solve and write interval notation for the solution set. Then graph the solution set. $$|2 x+3| \leq 9$$

Fill in the blank with the correct term. Some of the given choices will not be used. distance formula, midpoint formula, function, relation, \(x\) -intercept, y-intercept, perpendicular, parallel ,horizontal lines, vertical lines,symmetric with respect to the \(x\) -axis, symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, increasing, decreasing, constant The ___ is \(d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)

Solve. $$\sqrt{2 y-5}-\sqrt{y-3}=1$$
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