Problem 2 - Intermediate Algebra

Question

Simplify each numerical expression. \(2^{-4}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{16}\).

Step 1: Understanding the Problem

We need to simplify the expression \(2^{-4}\). This involves dealing with a negative exponent.

Step 2: Applying the Negative Exponent Rule

Recall the rule for negative exponents: for any non-zero number \(a\), \(a^{-n} = \frac{1}{a^n}\). Apply this rule to \(2^{-4}\) to rewrite it as \(\frac{1}{2^4}\).

Step 3: Calculating the Power of 2

Next, calculate \(2^4\). This means multiplying 2 by itself 4 times: \(2 \times 2 \times 2 \times 2 = 16\).

Step 4: Writing the Final Simplified Form

Now substitute the result of \(2^4\) back into the fraction: \(\frac{1}{2^4} = \frac{1}{16}\). This is the simplified form of the expression.

Key Concepts

Exponent Rules

Exponent rules are essential for simplifying expressions, especially when dealing with powers or roots of numbers. When we see an expression like \(2^{-4}\), we use the negative exponent rule. This specific rule states that for any non-zero base \(a\) raised to a negative exponent \(-n\), the expression is equal to the reciprocal of the base raised to the positive exponent:

• \(a^{-n} = \frac{1}{a^n}\)

This means we're essentially "flipping" the base to the denominator of a fraction.

This rule helps simplify negative exponents by making them positive, which allows for easier calculation. Remember, the exponent only affects the base itself, not any other part of an equation or expression.

Simplifying Expressions

When simplifying expressions such as \(2^{-4}\), you break the process into smaller, more manageable steps. Understanding the negative exponent rule is the first critical step. After that, you need to calculate the base number raised to the power indicated by the positive exponent, as in \(2^4\). This will give us a straightforward multiplication process.

• Identify the exponent and base.
• Use the negative exponent rule to transform the expression.
• Calculate the power of the base to simplify further.

Simplifying helps in reducing complex expressions into a simpler form, making them easier to work with in further mathematical operations. The goal is to reach a final, simplified expression that is both clear and concise.

Powers of Numbers

Calculating the powers of numbers can initially seem daunting, but it's a repetitive process of multiplying the base by itself. For the expression \(2^4\), you multiply the base (2) by itself four times:

• \(2 \times 2 = 4\)
• \(4 \times 2 = 8\)
• \(8 \times 2 = 16\)

Thus, \(2^4 = 16\).

Understanding this concept is crucial as it enables us to deal with more complex equations. Powers of numbers are a fundamental aspect of algebra and important for learning about exponential growth, scientific notation, and even logarithms. By mastering this skill, you can ease your journey into advanced mathematical concepts.