Problem 106 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ 2^{-5} $$

Step-by-Step Solution

Answer

Answer: \(\frac{1}{32}\)

Step 1: Write the expression as a fraction

First, we need to rewrite the expression with a negative exponent as a fraction. To do this, we can use the rule \(a^{-n} = \frac{1}{a^n}\). So, \(2^{-5}\) can be rewritten as: $$ 2^{-5} = \frac{1}{2^5} $$

Step 2: Calculate the denominator

Now, we need to find the value of \(2^5\). Remember that \(2^5\) means multiplying \(2\) by itself four more times: $$ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 $$

Step 3: Write the final answer

We found that the denominator is 32. The last step is to write the final answer. $$ \frac{1}{2^5} = \frac{1}{32} $$ So, the value of the expression \(2^{-5}\) is \(\frac{1}{32}\).

Key Concepts

Negative Exponents

Negative exponents might look a bit tricky at first, but understanding them is quite straightforward. When we have a negative exponent, like in the expression \(2^{-5}\), it essentially means that we are dealing with the reciprocal of the base raised to the corresponding positive exponent.

The rule to remember is:

• \(a^{-n} = \frac{1}{a^n}\)

This means \(2^{-5}\) is the same as \(\frac{1}{2^5}\). By using a negative exponent, we switch from multiplying the base number by itself, to dividing 1 by the number multiplied by itself. Understanding this rule will help simplify and solve many problems involving negative exponents.

Fraction Representation

Now, let's talk about converting expressions with negative exponents into fractions. As we've seen, the negative exponent indicates a reciprocal.

For example, in the case of \(2^{-5}\), converting it to a fraction involves flipping the position of 2, the base number, to the denominator:

• \(2^{-5} = \frac{1}{2^5}\)

Representing expressions as fractions is crucial in making calculations easier, especially when combining several expressions in math or performing further operations. This representation not only simplifies numerical evaluations but also reveals the relationship between different parts of the expression more clearly.

Numerical Expression Evaluation

Evaluating a numerical expression means calculating its value. When given an expression like \(2^{-5}\), we first rewrite using the rules of exponents and then evaluate its numerical value.

In our example, once we have \(2^{-5} = \frac{1}{2^5}\), we need to evaluate \(2^5\). This involves simple multiplication, \(2 \times 2 \times 2 \times 2 \times 2\).
This is equivalent to:

• 2 multiplied by itself 5 times
• Resulting in \(2^5 = 32\)

Thus, the expression \(2^{-5}\) evaluates to \(\frac{1}{32}\), giving us a clean and exact answer.

Powers of Two

The power of two is a fundamental concept in mathematics, computer science, and many fields of study. It denotes the number of times the base number 2 is multiplied by itself. So, for \(2^5\), this means multiplying 2, five times.

Understanding the power of two is particularly useful because it's a simple example, and powers of two often show up in computation and data storage.

• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)
• \(2^5 = 32\)

This sequence helps to understand the exponential growth that occurs with powers of two. Recognizing these patterns makes calculations involving powers much easier to tackle.