Problem 3 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 2^{-4} $$

Step-by-Step Solution

Answer

The value of \(2^{-4}\) is \(\frac{1}{16}\).

Step 1: Understand the Exponent

The expression given is \(2^{-4}\). The negative exponent indicates that the number should be taken as the reciprocal. So, we need to find the reciprocal of \(2^4\).

Step 2: Express with Positive Exponent

Convert \(2^{-4}\) to a fraction: \(\frac{1}{2^{4}}\). This means dividing 1 by \(2^4\).

Step 3: Calculate the Power

Calculate \(2^4\), which means multiplying 2 by itself 4 times: \(2 \times 2 \times 2 \times 2 = 16\).

Step 4: Find the Reciprocal

Now, find \(\frac{1}{2^4} = \frac{1}{16}\). This completes the evaluation of the expression.

Key Concepts

Reciprocals

Negative exponents might seem confusing at first, but they become straightforward once you understand the concept of reciprocals. The term reciprocal is often used in mathematics to mean the "flipping" of a fraction. More formally, the reciprocal of a number is 1 divided by that number.
When you see an expression like \(2^{-4}\), it indicates the reciprocal of \(2^4\). This is because the negative exponent suggests taking the inverse, or reciprocal, of the base raised to the positive power. So, in this case, \(2^{-4}\) becomes \(\frac{1}{2^4}\).
Key points to remember include:

• The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
• The reciprocal of a whole number \(x\) is \(\frac{1}{x}\).
• For any nonzero base with a negative exponent \(a^{-n}\), it equals \(\frac{1}{a^n}\).

Understanding reciprocals is essential for working with negative exponents and will make it easier to evaluate expressions involving them.

Positive Exponents

After determining the reciprocal, the next step in dealing with negative exponents is to express the exponent as positive. A positive exponent tells you how many times to multiply the base by itself.
Let’s take the expression \(2^{-4}\). By converting it to a positive exponent, we rewrite it as \(\frac{1}{2^4}\). Here, the burden is to deal only with \(2^4\), which simplifies calculations.
Positive exponents are straightforward due to the following attributes:

• They describe repeated multiplication, making \(a^n\) the same as multiplying \(a\) by itself \(n\) times.
• They are easy to compute once the expression is simplified (i.e., no negative signs involved in the exponent).

Additionally, remember that exponential growth occurs with positive exponents, reinforcing their practical importance.

Power Calculation

Finally, understanding power calculation is crucial when evaluating expressions with exponents. Calculating powers means multiplying the base by itself a given number of times, dictated by the exponent.
For our initial expression \(2^4\), this translates to multiplying 2 by itself four times, like so:

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

Thus, \(2^4 = 16\).
Power calculations follow basic multiplication and are fundamental in various algebraic expressions:

• Ensure to follow the order of operations, calculating powers before multiplication or division.
• Use power calculation for efficient solving of complex mathematical problems, especially when manual computation is required.

By mastering the techniques of power calculation, you'll improve your efficiency and accuracy in solving exponential expressions.