Problem 27 - Algebra for College Students

Question

Simplify each numerical expression. \(\left(4^{2} \cdot 5^{-1}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{256}{25}\).

Step 1: Apply the Power of a Power Rule

First, apply the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\). In this case, apply it to each part of the expression \((4^2 \cdot 5^{-1})^2\). Simplify the exponents inside the parentheses, which results in \((4^{2 \cdot 2} \cdot 5^{-1 \cdot 2}) = (4^4 \cdot 5^{-2})\).

Step 2: Simplify Each Term

Next, compute each term separately. Calculate \(4^4\), giving a result of 256, and \(5^{-2}\), which is equivalent to \(\frac{1}{5^2}\) or \(\frac{1}{25}\).

Step 3: Multiply the Results

Multiply the results of the previous step together. You have \(256 \cdot \frac{1}{25}\), which simplifies to \(\frac{256}{25}\).

Key Concepts

Power of a Power Rule

When you encounter expressions like \((a^m)^n\), they can look intimidating at first. But, there's a simple rule called the Power of a Power rule that can help simplify them. This rule states that you can multiply the exponents together: \((a^m)^n = a^{m \cdot n}\).
Let's break it down. Suppose you have \((4^2)^2\). According to this rule, you multiply the exponents, which are 2 and 2 in this case, giving you \(4^{2 \times 2} = 4^4\). It’s like repeatedly applying the base to itself, making it much simpler.
Remember, apply this rule to each part of the expression individually if there are multiple terms. For example, in \((4^2 \cdot 5^{-1})^2\), apply it to both 4 and 5 independently, resulting in \(4^4 \cdot 5^{-2}\). This step streamlines the process, making calculations easier in later steps.

Negative Exponents

Negative exponents might seem tricky, but they are simply the inverse of their positive counterparts. The key thing to remember is that a negative exponent indicates the reciprocal. For example, \(a^{-n} = \frac{1}{a^n}\).
So, if you stumble upon something like \(5^{-2}\), it transforms into the fraction \(\frac{1}{5^2}\), or \(\frac{1}{25}\). This is because the negative exponent signifies that we're flipping the base to the denominator.
Taking care of negative exponents early can simplify further calculations. If your expression involves both negative and positive exponents, like our example, manage them one at a time to avoid confusion. This approach keeps the math neat and errors at bay.

Multiplying Fractions

Multiplying fractions is a foundational skill that can simplify complex math problems. To multiply fractions:

• Multiply the numerators (the top numbers) together.
• Multiply the denominators (the bottom numbers) together.

These simple rules turn intimidating problems into manageable ones. For our example, after simplifying for exponents, you get a fraction like \(\frac{256}{1} \times \frac{1}{25}\).
Multiply straight across, top and bottom to get \(\frac{256 \times 1}{1 \times 25} = \frac{256}{25}\).
The result is already simplified if there are no common factors in the numerator and the denominator. Mastering this process makes dealing with fractions second nature and ensures your calculations are accurate.