Problem 48 - Beginning and Intermediate Algebra

Question

Evaluate. $$ -5^{2} $$

Step-by-Step Solution

Answer

The result is -25.

Step 1: Identify the Base and Exponent

The expression given is \(-5^2\).Here, 5 is the base, and 2 is the exponent.

Step 2: Apply the Order of Operations

According to the order of operations (PEMDAS/BODMAS), exponents are evaluated before any addition or subtraction. Therefore, we evaluate \(5^2\) first.

Step 3: Evaluate the Exponent

Calculate \(5^2\) by multiplying the base (5) by itself: \[5 \times 5 = 25\]

Step 4: Apply the Negative Sign

The negative sign is not included in the exponentiation process because it only applies to the 5. Thus, after calculating the exponent, apply the negative sign to the result: \[-25\]

Step 5: Conclusion: Evaluate the Expression

After following the steps, the value of the expression \(-5^2\) is -25.

Key Concepts

Order of Operations

When solving mathematical problems, it's crucial to follow the order in which operations are performed. The order of operations, often remembered by the acronym PEMDAS or BODMAS, provides a guideline to correctly evaluate expressions. This order is as follows:

• P/B - Parentheses/Brackets
• E/O - Exponents/Orders
• MD - Multiplication and Division (left to right)
• AS - Addition and Subtraction (left to right)

By adhering to this sequence, especially focusing on exponents early in the process, you ensure that you arrive at the correct result. Always remember to evaluate any exponents present before moving on to other operations. In the case of the problem at hand \(-5^2\), notice that the exponentiation occurs before the application of the negative sign. This can sometimes be a source of confusion, but understanding that the exponent has priority in the operations helps avoid mistakes.

Base and Exponent

In expressions involving exponents, recognizing the base and exponent plays a pivotal role. Here, the base is the number that is being multiplied by itself, and the exponent tells you how many times the base is used as a factor. For example, in the expression \(5^2\), 5 is the base and 2 is the exponent, indicating that 5 should be multiplied by itself once, leading to \(5 \times 5\).
This concept becomes more interesting when negative signs are involved. The negative sign is often attached to the base and does not affect the exponentiation process directly. Thus, in the expression \(-5^2\), the exponent 2 applies only to 5, not the negative sign outside it. Consequently, you first calculate the exponent \(5^2 = 25\), and then apply the negative sign resulting in a final value of -25. This clarifies the common mistakes where students think the entire expression includes the negative base being squared, which would be written as \((-5)^2\).

Evaluate Expressions

Evaluating expressions correctly is a fundamental skill in mathematics. It involves applying key principles, such as the order of operations and understanding how bases and exponents work, to simplify an expression to its simplest form. In our problem, \(-5^2\), the evaluation involves taking careful steps.

• Firstly, identify the parts of the expression: the base 5 and the exponent 2.
• Next, compute the exponent: \(5^2 = 25\).
• Finally, apply any additional operators, such as the negative sign: yielding -25.

This structured approach ensures you are following a logical sequence, leading to the correct answer. Always take care when you encounter negative signs or other operations outside the scope of exponentiation. By doing so, you maintain clarity and accuracy in your mathematical calculations. As you practice more, evaluating expressions will become second nature, helping you solve more complex problems with ease.