Problem 6 - Algebra and Trigonometry

Question

Express the following numbers without using exponents. (a) \(2^{-1}=\) _____. (b) \(2^{-3}=\) _____. (c) \(\left(\frac{1}{2}\right)^{-1}=\) _____.

Step-by-Step Solution

Answer

(a) \(\frac{1}{2}\), (b) \(\frac{1}{8}\), (c) 2.

Step 1: Understanding the Negative Exponent

A negative exponent indicates the reciprocal of that base raised to the positive of the exponent. For example, if you have a number with an exponent of \(-n\), it means \(1\) divided by that number raised to the \(n\). For instance, \(2^{-1}\) becomes \(\frac{1}{2^1}\).

Step 2: Solving Part (a)

Using the rule from Step 1, we convert \(2^{-1}\) to \(\frac{1}{2^1}\). Since \(2^1 = 2\), we have \(\frac{1}{2}\). Therefore, \(2^{-1} = \frac{1}{2}\).

Step 3: Solving Part (b)

Similarly for part (b), \(2^{-3}\) becomes \(\frac{1}{2^3}\). Calculating \(2^3\), which is \(2 \times 2 \times 2 = 8\), gives \(\frac{1}{8}\). So, \(2^{-3} = \frac{1}{8}\).

Step 4: Solving Part (c)

The expression \(\left(\frac{1}{2}\right)^{-1}\) means to take the reciprocal of \(\frac{1}{2}\), which is \(\frac{1}{\frac{1}{2}}\). Taking the reciprocal gives \(2\). Hence, \(\left(\frac{1}{2}\right)^{-1} = 2\).

Key Concepts

Understanding Reciprocals

The idea of a reciprocal is essential in mathematics, especially when dealing with negative exponents. A reciprocal of a number is simply 1 divided by that number. Think of it as flipping fractions upside down.
For instance:

• The reciprocal of 2 is \( \frac{1}{2} \).
• The reciprocal of \( \frac{1}{3} \) is 3. This is because when you divide 1 by \( \frac{1}{3} \), you essentially multiply by the inverse, hence, \( 1 \times \frac{3}{1} = 3 \).
• Any whole number "n" is technically n/1, so its reciprocal is \( \frac{1}{n} \).

Understanding how to find reciprocals is crucial when dealing with negative exponents. As you'll see later, a negative exponent converts the base to its reciprocal.

Exploring Base and Exponent Relationships

In mathematics, an exponent represents how many times a number, known as the base, is multiplied by itself. For example, \( 2^3 \) means multiplying 2 by itself three times: \( 2 \times 2 \times 2 = 8 \).
But what happens when you encounter a negative exponent like in \( 2^{-3} \)?

• Instead of multiplying, a negative exponent means dividing 1 by the base raised to the corresponding positive exponent:
  • \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \).
• Another important point is when the base is a fraction, as in \( \left(\frac{1}{2}\right)^{-1} \):
  • Taking it to the power of \(-1\) flips the fraction to give the reciprocal, thus \( \frac{1}{2} \) becomes 2.

The base-exponent relationship helps in breaking down complex expressions, especially when you combine multiplication, division, and powers.

Simplifying Fractions with Negative Exponents

The process of fraction simplification comes in handy when working with negative exponents. When you multiply or divide fractions, it's important to simplify wherever possible to make calculations easier. Let's solve and simplify some problems:

• In the exercise, converting \( 2^{-1} \) involves turning \( 2 \) into \( \frac{1}{2} \), since \( 2^{-1} = \frac{1}{2^1} \), thus simplifying directly to \( \frac{1}{2} \).
• For \( 2^{-3} = \frac{1}{8} \), the operation was simplified further by breaking it down: \( 2^3 = 8 \), then expressing it as \( \frac{1}{8} \).
• When simplifying \( \left(\frac{1}{2}\right)^{-1} \), you switch the numerator and denominator to get back to 2 as the simplification.

By mastering fraction simplification, especially with negative exponents, you can handle various mathematical expressions accurately and efficiently.