Problem 24 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ \frac{1}{4^{-2}} $$

Step-by-Step Solution

Answer

The expression evaluates to 16.

Step 1: Understand the Problem

The given expression is \( \frac{1}{4^{-2}} \). Our task is to simplify this expression by manually evaluating it.

Step 2: Apply the Negative Exponent Rule

Recall the negative exponent rule: \( a^{-n} = \frac{1}{a^n} \). Applying this to the denominator, \( 4^{-2} = \frac{1}{4^2} \). So the expression becomes \( \frac{1}{\frac{1}{4^2}} \).

Step 3: Simplify the Fraction

When dividing by a fraction, you multiply by its reciprocal. So, \( \frac{1}{\frac{1}{4^2}} = 4^2 \).

Step 4: Calculate the Power of 4

Calculate \( 4^2 \), which is \( 4 \times 4 = 16 \). So the expression simplifies to \( 16 \).

Step 5: Confirm the Result with a Calculator

Using a calculator, input \( \frac{1}{4^{-2}} \) to confirm the result. The calculator should give \( 16 \) as the solution, confirming our manual calculation.

Key Concepts

Simplifying Expressions

Simplifying expressions, especially those with negative exponents, can seem daunting at first. But it is really just about understanding a few basic rules. For the expression \( \frac{1}{4^{-2}} \), it includes a negative exponent in the denominator. To simplify this, we apply the negative exponent rule: \( a^{-n} = \frac{1}{a^n} \).
This rule helps to transform the negative exponent into a manageable positive exponent. So, \( 4^{-2} \) in the denominator becomes \( \frac{1}{4^2} \). Now the task is to simplify the resulting complex fraction, \( \frac{1}{\frac{1}{4^2}} \). Remember, when you have a fraction under a fraction, it usually simplifies to just the numerator of the fraction.
So, \( \frac{1}{\frac{1}{4^2}} = 4^2 \). This rule makes dealing with exponents less intimidating and simplifies expressions efficiently.

Calculator Verification

After simplifying expressions manually, it is always a smart move to verify your results with a calculator. This ensures that you have executed the steps correctly. For our expression \( \frac{1}{4^{-2}} \), after simplification we ended up with \( 16 \). To verify this result, you can use a scientific calculator.
Input the original expression exactly as it is: \( \frac{1}{4^{-2}} \). Ensure that you open and close parentheses appropriately, as scientific calculators can interpret inputs differently depending on their sequence.
On calculating, the display should show \( 16 \), confirming your manual result. This step reinforces your understanding and gives you confidence that you've simplified the expression correctly.

Fraction Operations

Operations with fractions form a crucial part of simplifying expressions. Specifically, dealing with division involving fractions can be tricky until you master the rules. In the expression \( \frac{1}{\frac{1}{4^2}} \), you encounter a division of fractions, which is handled by multiplying by the reciprocal.
The reciprocal of a fraction \( \frac{1}{4^2} \) is simply flipping the numerator and the denominator, resulting in \( 4^2 \). So you multiply the outer fraction, \( 1 \), by this reciprocal to simply get \( 4^2 \). This transforms complex-looking problems into basic multiplication problems that are easy to compute.
Fraction operations become much more straightforward with this approach, allowing you to tackle complex expressions with confidence.