Problem 68 - Algebra 1: Concepts and Skills

Question

Which expression equals \(\frac{1}{8} ?\) A. \(-8\) B. \(4^{-2}\) C. \(2^{-3}\) D. \(1^{-8}\)

Step-by-Step Solution

Answer

The correct expression that equals \(\frac{1}{8}\) is C. \(2^{-3}\).

Step 1: Evaluate Expression A

Evaluate expression A: \(-8\). The result is \(-8\), which is not equal to \(\frac{1}{8}\). This means A is not the correct answer.

Step 2: Evaluate Expression B

Evaluate expression B: \(4^{-2}\). A negative exponent essentially means taking the reciprocal of the base raised to that exponent. \(4^{-2}\) equals \( \frac{1}{4^2}\), or \(\frac{1}{16}\). So, B also does not equal \(\frac{1}{8}\), and is not the correct answer

Step 3: Evaluate Expression C

Evaluate expression C: \(2^{-3}\). \(2^{-3}\) equals \(\frac{1}{2^3}\), or \(\frac{1}{8}\). Therefore, C is equal to \(\frac{1}{8}\), and is the correct answer

Step 4: Check Expression D

Evaluate expression D: \(1^{-8}\). Any non-zero number raised to any power equals to one, so \(1^{-8}\) is also 1. D is not the correct answer, but the check confirms that C is indeed the correct answer.

Key Concepts

Negative Exponents

When you see a negative exponent, it might seem confusing at first. However, it's just a shorthand way to write fractions or reciprocals. A negative exponent means you take the reciprocal of the base raised to the opposite positive exponent.

For example, if you have an expression like \(a^{-b}\), you can convert it to a fraction by writing it as \(\frac{1}{a^b}\). This is a very useful rule because it allows us to simplify expressions and solve problems more easily. In the context of evaluating expressions, understanding negative exponents helps you rewrite and simplify them quickly, as in the problem where \(2^{-3}\) is simplified to \(\frac{1}{8}\).

Remember:

• Negative exponent = Reciprocal
• \(a^{-b} = \frac{1}{a^b}\)

Reciprocal

A reciprocal is simply one divided by the number in question. If you have a number \(a\), its reciprocal is \(\frac{1}{a}\). The concept of reciprocal is important in mathematics, especially when dealing with fractions and negative exponents.

Let's clarify with an example. The reciprocal of 4 is \(\frac{1}{4}\), and likewise, the reciprocal of \(8\) is \(\frac{1}{8}\). In our problem, by understanding the reciprocal concept, we easily saw that \(2^{-3}\) converted to its reciprocal form as \(\frac{1}{8}\), matching the given expression.

Key points to remember:

• Reciprocal of \(a\) is \(\frac{1}{a}\)
• Used often with negative exponents: it helps to rewrite and simplify expressions

Evaluating Expressions

Evaluating expressions is all about simplifying them to find their value. It involves understanding the mathematical principles and operations within an expression.

For the given exercise, evaluating each option correctly led us to the right answer. Here's a quick breakdown:

• Check each expression's result: Calculate and simplify each to find if it equals the target value.
• Use the key concepts: Knowledge of negative exponents and reciprocals simplifies the task.
• Verify your results: By checking all possibilities, like in the steps provided, you can ensure correctness.

This method of evaluating helps identify the expression that correctly equals \(\frac{1}{8}\) from the options given, i.e., \(2^{-3}\). Remember always to perform each calculation carefully and utilize your understanding of underlying mathematical concepts.