Problem 23 - College Algebra

Question

Simplify the given expression. $$ 25 \div 5^{2}-7 $$

Step-by-Step Solution

Answer

The simplified expression is -6.

Step 1: Evaluate the Exponent

First, calculate the exponent in the expression: \[ 5^2 = 5 \times 5 = 25 \]

Step 2: Perform the Division

Now use the result from Step 1 to evaluate the division part of the expression:\[ 25 \div 25 = 1 \]

Step 3: Perform the Subtraction

Finally, subtract 7 from the result obtained in Step 2:\[ 1 - 7 = -6 \]

Key Concepts

Exponents

Exponents are a fundamental part of algebra simplification. They show how many times a number, called the base, is multiplied by itself. In the expression, we have the base 5 raised to the power of 2, written as \(5^2\). This means we multiply 5 by itself once:

• \(5 \times 5 = 25\)

Knowing how to calculate exponents is crucial when simplifying expressions because they often appear in mathematical problems. Calculating exponents correctly ensures that other operations based on them, like division and subtraction, are accurate.

Division

Division is the process of finding how many times one number is contained in another. In this expression, we divide 25 by the result of the exponent, 25. Division here can be thought of as distributing apples equally between friends - you are deciding how many apples each friend will get. For our example, since both numbers are 25, it simplifies accordingly:

• \(25 \div 25 = 1\)

Understanding how division interacts with other operators like exponents simplifies complex expressions into easier numbers to work with.

Order of Operations

Order of operations is a rule that defines the sequence in which operations should be performed to ensure a single, correct result. This order is often remembered by the acronym PEMDAS:

• P: Parentheses
• E: Exponents
• M/D: Multiplication/Division (left to right)
• A/S: Addition/Subtraction (left to right)

In our exercise, the order of operations guided us to first evaluate the exponent \(5^2\), followed by the division \(25 \div 25\), and finally the subtraction \(1 - 7\). It's important to follow this order to avoid mistakes and get the correct result, especially when multiple operations are involved.

Subtraction

Subtraction is the process of taking one number away from another. It's like removing apples from a basket. After performing operations like exponents and division, subtraction often helps find the final simplified result of an expression. In the given expression, after determining the division result:

• Start from the answer of the division: \(1\)
• Subtract 7 from 1: \(1 - 7 = -6\)

Subtraction requires careful attention, especially when working with negative results or larger numbers. This ensures that the final outcome reflects the correct adjustment.