Problem 15 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ \left(0.5^{-2}\right)^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 16.

Step 1: Evaluate the Inner Exponentiation

First, evaluate the expression inside the parentheses: \(0.5^{-2}\). This means we need to find the reciprocal of 0.5 raised to the power of 2.Reciprocal of 0.5 is \(1/0.5 = 2\). Hence, \(0.5^{-2} = 2^2\).

Step 2: Calculate the Result of 2 Squared

Now, solve \(2^2\), which is the result from Step 1. This is a straightforward calculation: \(2^2 = 4\).

Step 3: Evaluate the Outer Exponentiation

Take the result from Step 2, which is \(4\), and raise it to the power of 2, as indicated by the original outer exponentiation.Calculate \(4^2\): \(4^2 = 16\).

Key Concepts

Reciprocal

When we talk about the reciprocal of a number, we're referring to the value that, when multiplied by the original number, will give a product of 1. For any non-zero number, its reciprocal is simply one divided by that number.
Understanding reciprocals is especially important when dealing with negative exponents.

• A negative exponent means that the base should be inverted to its reciprocal form and then raised to the positive power of that exponent.
• For instance, the expression \(0.5^{-2}\) is equivalent to the reciprocal of 0.5 raised to the power of 2.

For the value 0.5, its reciprocal is calculated as \(1 \div 0.5\), which actually equals 2.
This understanding allows us to transform \(0.5^{-2}\) into \(2^2\), simplifying the calculation.

Inner Exponentiation

Inner exponentiation refers to the process of first dealing with any exponents inside parentheses. This is like tackling the most internal part of an expression that involves exponents.
Here's how to handle it:

• Identify terms inside the parentheses that have exponents.
• Evaluate these terms from the inside-out, paying special attention to any negative exponents.

In the expression \(\left( 0.5^{-2} \right)^2\), the inner exponentiation step focuses initially on \(0.5^{-2}\).
By first finding the reciprocal of 0.5 and taking it to the power of 2, we simplify the expression to \(2^2\). Once this is resolved, the result acts as the base for the next layer of exponentiation.

Outer Exponentiation

Outer exponentiation comes into play after the inner exponentiation has been resolved. This step involves applying any remaining exponents that are outside of the parentheses.
Essentially, it is the final touch to complete the evaluation of the expression.

• Begin by taking the result from the inner exponentiation.
• Raise this result to the power indicated by the outer exponentiation.

In the original problem \(\left(0.5^{-2}\right)^{2}\), after simplifying the inner part to \(2^2 = 4\), we treat 4 as the new base for outer exponentiation.
We then raise 4 to the power of 2, completing the calculation: \(4^2 = 16\). This final step ensures that the entire expression has been correctly evaluated.