Problem 59 - Elementary Algebra

Question

Find the value of \(4^{-2} \cdot 8^{2}-3^{2}\).

Step-by-Step Solution

Answer

Answer: -5

Step 1: Solve exponent terms

To start, we need to find the values of \(4^{-2}\), \(8^{2}\) and \(3^{2}\). We use exponent rules to solve each term: 1. For \(4^{-2}\): Since the exponent is negative, we convert it into positive by taking the reciprocal of the base: \((4^{-2}) = (\frac{1}{4})^{2}\) Now, square the fraction: \((\frac{1}{4})^{2} = \frac{1^2}{4^2} = \frac{1}{16}\) 2. For \(8^{2}\): Square the base 8: \((8^{2}) = 8 \cdot 8 = 64\) 3. For \(3^{2}\): Square the base 3: \((3^{2}) = 3 \cdot 3 = 9\)

Step 2: Perform multiplication

Now, multiply the values of \(4^{-2}\) and \(8^{2}\): \(\frac{1}{16} \cdot 64 = \frac{64}{16} = 4\)

Step 3: Perform subtraction

Finally, subtract the value of \(3^{2}\) from the result of the multiplication in the previous step: \(4 - 9 = -5\) So, the value of the given expression \(4^{-2} \cdot 8^{2}-3^{2} = -5\).

Key Concepts

Negative Exponents

A negative exponent indicates that we should take the reciprocal of the base and then apply the exponent. For example, if we have a term like \(4^{-2}\), we change the base from a whole number to a fraction. This is done because negative exponents "flip" the number. The definition goes as follows:
\[a^{-n} = \frac{1}{a^n}\]This means that \(4^{-2}\) becomes \(\left(\frac{1}{4}\right)^2\). After flipping, apply the exponent to the fraction which equates to squaring it.
Squaring both the numerator and the denominator, we get:

• Numerator: \(1^2 = 1\)
• Denominator: \(4^2 = 16\)

So, the final result is \(\frac{1}{16}\). In summary, negative exponents are useful for transforming values into fractions, which can then be handled with regular exponent rules.

Fractional Exponents

Fractional exponents appear in expressions when dealing with roots. However, in this exercise, we keep the concept straightforward by working with whole numbers after resolving the negative exponent. Fractional exponents express power in the form of a fraction, but in our case, they represent multiplicative operations rather than roots.
It's essential to know that \(a^{\frac{m}{n}}\) represents the \(n\)-th root of \(a^m\).For instance, often you'll see equations like \((a^{\frac{1}{2}}) = \sqrt{a}\), illustrating a square root. Yet our problem simplifies this; \(a^{\frac{m}{n}}\) is not directly used but still strongly ties into simplifying complex exponents in their full forms.
Fractional exponents help bridge the gap between multiplication and roots in algebra, but focus on simplification first if no root is required.

Order of Operations

Order of operations is a fundamental principle that dictates how we solve expressions step-by-step. It's important to remember the acronym PEMDAS:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

This order ensures that calculations are performed consistently and correctly. In our expression \(4^{-2} \cdot 8^{2} - 3^{2}\), we tackle the exponents first. That means resolving each part:

• First, solve the terms \(4^{-2}\), \(8^{2}\), and \(3^{2}\).
• Next, perform the multiplication \(\frac{1}{16} \cdot 64\).
• Finally, complete the subtraction step, subtracting 9 from 4.

The sequence in solving these operations determines your result, ensuring accurate calculations and avoiding mistakes.