Problem 3 - Precalculus: Functions and Graphs

Question

Express the number in the form \(a / b,\) where \(a\) and \(b\) are integers. $$\frac{2^{-3}}{3^{-2}}$$

Step-by-Step Solution

Answer

The fraction is \( \frac{9}{8} \).

Step 1: Understand Negative Exponents

Negative exponents represent the reciprocal of the positive exponent. Specifically, the expression \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \). Therefore, \( 2^{-3} = \frac{1}{2^3} \) and \( 3^{-2} = \frac{1}{3^2} \).

Step 2: Calculate Positive Exponents

Now calculate \( 2^3 \) and \( 3^2 \). We get \( 2^3 = 8 \) and \( 3^2 = 9 \). Thus, \( 2^{-3} = \frac{1}{8} \) and \( 3^{-2} = \frac{1}{9} \).

Step 3: Divide Fractions by Multiplying

The expression \( \frac{2^{-3}}{3^{-2}} \) is equal to \( \frac{\frac{1}{8}}{\frac{1}{9}} \). To divide fractions, multiply by the reciprocal of the divisor: \( \frac{1}{8} \times 9 = \frac{9}{8} \).

Step 4: Simplify If Needed

The fraction \( \frac{9}{8} \) is already in its simplest form, as 9 and 8 have no common factors other than 1.

Key Concepts

Reciprocal

In mathematics, the concept of a reciprocal is quite fundamental. A reciprocal essentially means flipping a number. For instance, the reciprocal of a number \( x \) is \( \frac{1}{x} \). This is easy to visualize if you think about multiplying a number by its reciprocal, which always gives 1. This is because \( x \times \frac{1}{x} = 1 \).

When we deal with negative exponents, understanding reciprocals becomes important. If you encounter \( a^{-n} \), it means you are looking at the reciprocal of \( a^n \). So, \( 2^{-3} \) becomes \( \frac{1}{2^3} \), and \( 3^{-2} \) becomes \( \frac{1}{3^2} \). This reciprocal operation allows negative exponents to turn into positive exponents by simply flipping the number involved.

Power of Integers

When dealing with powers, or exponents, you are expressing how many times to multiply a number by itself. For example, \( 2^3 \) means \( 2 \times 2 \times 2 = 8 \). The number \( 2 \) here is the base, and \( 3 \) is the exponent, which tells us the power of the base.

For negative exponents, we first turn them into their positive form using reciprocals. This then lets us calculate the power for the original base. As we saw earlier:

• \( 2^{-3} = \frac{1}{2^3} \)
• \( 2^3 = 8 \)
• Therefore, \( 2^{-3} = \frac{1}{8} \)

Similarly, \( 3^{-2} \) turns into \( \frac{1}{3^2} \), which is \( \frac{1}{9} \) because \( 3^2 = 9 \). By understanding powers of integers, you can easily work with both positive and negative exponents.

Fraction Division

Dividing fractions might initially seem tricky, but it's all about using reciprocals to make it easier. The rule for division of fractions is to multiply by the reciprocal. If you have a division situation like \( \frac{a}{b} \div \frac{c}{d} \), it becomes \( \frac{a}{b} \times \frac{d}{c} \).

In the problem \( \frac{2^{-3}}{3^{-2}} \), we calculated these as \( \frac{1}{8} \) and \( \frac{1}{9} \) respectively. To divide these, you multiply by the reciprocal of the second fraction:

• \( \frac{1}{8} \div \frac{1}{9} \rightarrow \frac{1}{8} \times 9 \)
• The result is \( \frac{9}{8} \)

This process works because multiplying by a reciprocal effectively reverses the division, simplifying the fraction operation.