Problem 98 - Precalculus Student Solutions Manual 5th

Question

Explain why the following method of simplifying \(i^{-42}\) works. $$ i^{-42}=\frac{1}{i^{42}}=\frac{1}{\left(i^{2}\right)^{21}}=\frac{1}{(-1)^{21}}=\frac{1}{-1}=-1 $$

Step-by-Step Solution

Answer

Simplifying results in \( -1 \).

Step 1: Expressing the negative exponent

Rewrite the expression with a positive exponent using the property of negative exponents: \( i^{-42} = \frac{1}{i^{42}} \)

Step 2: Breaking down the exponent

Express the exponent \(42\) as a product of 2 and 21: \( i^{42} = (i^2)^{21} \)

Step 3: Simplifying using powers of i

Recognize that \(i^2 = -1\), then substitute this value into \((i^2)^{21}\): \(i^{42} = (-1)^{21} \)

Step 4: Calculating the power of -1

Simplify \((-1)^{21}\). Since \(21\) is an odd number, \((-1)^{21} = -1\): \( \frac{1}{(-1)^{21}} = \frac{1}{-1} \)

Step 5: Final simplification

Finally, simplify: \( \frac{1}{-1} = -1 \)

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they’re quite simple once you understand the basic idea. When you see a negative exponent, it means you take the reciprocal of the base raised to the corresponding positive exponent. For example, if you have a negative exponent like this: \( a^{-n} \), you can rewrite it as \( \frac{1}{a^n} \). In our specific problem, we start with \( i^{-42} \). Using the property of negative exponents, we rewrite it as \( \frac{1}{i^{42}} \). By doing this, the problem becomes much easier to handle.

Powers of Imaginary Unit

Understanding the powers of the imaginary unit \(i\) is crucial. Remember that \(i\) is defined as the square root of -1 (\(i = \sqrt{-1} \)). Because of this, \(i^2 = -1\). The powers of \(i\) repeat in a cycle of four:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

Knowing this cycle helps to simplify higher powers of \(i\). For instance, \(i^{42}\) can be broken down to see how it fits into this cycle: \(i^{42} = (i^2)^{21}\). Since \(i^2 = -1\), the problem simplifies to \(i^{42} = (-1)^{21}\) which is easier to evaluate.

Simplifying Expressions

Simplifying expressions involves using algebraic rules and properties to make an expression easier to work with. In our example, we move from \(i^{-42}\) to \(\frac{1}{i^{42}}\) using the property of negative exponents. Next, we recognize that \(i^{42} = (i^2)^{21}\), which uses the exponent multiplication rule: \((a^m)^n = a^{mn}\). By recognizing that \(i^2 = -1\), we can replace \((i^2)^{21}\) with \((-1)^{21}\), greatly simplifying our expression. Finally, knowing that an odd power of -1 is -1, we see that \((-1)^{21} = -1\). Therefore, our simplified form is \(\frac{1}{-1} = -1\).

Properties of Exponents

Several properties of exponents come into play when solving problems with complex numbers and negative exponents. Here are the key properties used in our example:

• Negative exponent rule: \(a^{-n} = \frac{1}{a^n}\).
• Exponent multiplication rule: \((a^m)^n = a^{mn}\).

We apply the negative exponent rule to change \(i^{-42}\) to \(\frac{1}{i^{42}}\). Then we use the exponent multiplication rule to break down \(i^{42}\) into \((i^2)^{21}\). This lets us replace \(i^2\) with -1. Understanding and applying these properties is vital for working with exponents, whether they’re positive, negative, or involve complex numbers. By mastering these rules, you can simplify and solve expressions more effectively.