Problem 26 - College Algebra

Question

Simplify the given expression. $$ 4^{2}-25 \cdot \frac{1}{5} $$

Step-by-Step Solution

Answer

The simplified expression is 11.

Step 1: Square the Base

Begin by evaluating the expression involving exponentiation. The given expression is \(4^2\). To simplify, calculate \(4^2 = 4 \times 4 = 16\).

Step 2: Multiply Inside the Fraction

Next, simplify the multiplication inside the fraction. The expression is \(25 \cdot \frac{1}{5}\). This can be calculated as: \(25 \times \frac{1}{5} = \frac{25}{5} = 5\).

Step 3: Subtract the Result

Finally, subtract the result from Step 2 from the result of Step 1. That is, calculate \(16 - 5 = 11\).

Key Concepts

Exponents

Exponents are powerful mathematical tools that tell us how many times to multiply a number by itself. When working with exponents, understanding the base and the exponent is crucial. In our example, the base is 4, and the exponent is 2, giving us the expression \(4^2\). This means we should multiply 4 by itself, resulting in \(4 \times 4 = 16\).

Working with exponents can simplify expressions significantly, especially in cases of large numbers, because they compactly represent repeated multiplication. Always perform the exponentiation early in the process when simplifying expressions, as it can influence how the rest of the expression is calculated.

Multiplication

Multiplication is a basic arithmetic operation that combines groups of equal sizes. In algebra, it often involves multiplying whole numbers, fractions, or even more complex entities like variables. In this exercise, we face a multiplication situation involving a whole number and a fraction: \(25 \cdot \frac{1}{5}\).

The multiplication of a fraction by a whole number is straightforward. Here, we multiply 25 by \(\frac{1}{5}\), which can be rewritten as \(\frac{25}{5}\). Simplifying this fraction by dividing 25 by 5 gives us 5. Remember to always simplify fractions to their lowest terms to make calculations easier and to get the clearest result possible.

Subtraction

Subtraction is the process of taking one quantity away from another. It is essential when combining terms to obtain the simplest form of an expression. In our scenario, subtraction comes into play in the final step, where we subtract the result of one operation from another. Specifically, after calculating the exponent to obtain 16 and the multiplication to get 5, we perform the subtraction: \(16 - 5\).

The result is 11, which is the simplified form of the original expression. When subtracting, ensure all terms are calculated correctly to prevent errors. Subtraction can change the expression's value significantly, so it is often considered the final, crucial step in simplifying mathematical expressions.