Problem 111 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ 2^{-3}\left(3^{-2}\right) $$

Step-by-Step Solution

Answer

Answer: \(\frac{1}{72}\)

Step 1: Simplify the first exponent

First, let's simplify \(2^{-3}\). A negative exponent means to take the reciprocal of the base raised to the exponent's positive counterpart. So, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).

Step 2: Simplify the second exponent

Now, let's simplify \(3^{-2}\). Again, a negative exponent means to take the reciprocal of the base raised to the exponent's positive counterpart. So, \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).

Step 3: Multiply the simplified expressions

We now have \(\frac{1}{8}\left(\frac{1}{9}\right)\). To multiply fractions, multiply the numerators together and the denominators together. This gives us \(\frac{1\cdot1}{8\cdot9} = \frac{1}{72}\). The final answer for the given numerical expression is \(\frac{1}{72}\).

Key Concepts

Understanding Reciprocals

When we talk about the reciprocal, we are essentially talking about flipping a fraction upside down. For example, the reciprocal of a simple fraction like \(\frac{3}{4}\) is \(\frac{4}{3}\). The reciprocal of a number is what you multiply it by to get 1. It's a key concept when dealing with negative exponents.

With negative exponents, the idea is straightforward: they signal that we should take the reciprocal of the base raised to the corresponding positive exponent. So, for \(2^{-3}\), the negative exponent tells us to flip the base of 2 upside down and raise it to the positive 3. This gives us \(\frac{1}{2^3}\).
Similarly, for \(3^{-2}\), it becomes \(\frac{1}{3^2}\). Keep in mind: converting a negative exponent to its reciprocal form simplifies expressions and helps with further operations like multiplication.

Multiplying Fractions Simplified

Multiplying fractions might seem complicated, but it's actually very straightforward. When multiplying two fractions, each with a numerator and a denominator, your task is to multiply the numerators together, and then multiply the denominators together.

For example, if you have \(\frac{a}{b} \times \frac{c}{d}\), you multiply \(a\times c\) to get the new numerator, and \(b\times d\) to get the new denominator. This results in a new fraction: \(\frac{a \times c}{b \times d}\).
In our exercise, we multiplied \(\frac{1}{8}\) by \(\frac{1}{9}\). This means:

• Numerators: \(1 \times 1 = 1\)


• Denominators: \(8 \times 9 = 72\)

Therefore, \(\frac{1}{72}\) becomes our answer. Simplifying and multiplying fractions becomes much less daunting with these basic rules in mind.

Evaluating Numerical Expressions

Evaluating a numerical expression means calculating its value. It often involves a series of operations, like simplifying exponents or multiplying fractions, to reach the final result.

To evaluate expressions involving negative exponents, we use the rule of reciprocals to convert them into fractions. Once we've converted these components into simpler fractions, we look at how they interact through multiplication. By smoothly handling each step separately—simplifying exponents first, and then moving on to multiplication—we ensure that our calculations are accurate and easy to follow.

For instance, in the exercise \(2^{-3}(3^{-2})\), we broke it down to \(\frac{1}{8} \times \frac{1}{9}\). Following the multiplication process, it reduces down to \(\frac{1}{72}\). Breaking it down in these ways helps us see how each mathematical operation builds on the others to give us the final, correct answer.