Problem 49 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The expression \((-5)^2\) evaluates to 25.

Step 1: Understanding the Problem

The problem requires us to evaluate the expression \((-5)^2\). This involves raising the base, which is \(-5\), to the power of 2. We must remember that raising a number to the power of 2 means multiplying the number by itself.

Step 2: Applying the Power Rule

To solve \((-5)^2\), we use the rule that \(a^b = a \times a\times \, \ldots \text{ (b times)}\). For our problem, this means we have \(-5\) multiplied by itself exactly two times. So, \((-5)^2 = (-5) \times (-5)\).

Step 3: Multiplying Negative Numbers

Recall the rule for multiplying negative numbers: the product of two negative numbers is positive. Therefore, when we calculate \((-5) \times (-5)\), we multiply the absolute values and the result is positive. Thus, \((-5) \times (-5) = 25\).

Step 4: Final Evaluation

After calculating, we find that \((-5)^2 = 25\). Thus, the expression evaluates to 25.

Key Concepts

Understanding the Power of a Number

The concept of raising a number to a power is quite intuitive once you break it down. When we say a number is raised to a power, we're essentially saying to multiply that number by itself a certain number of times. For example,

• The base is the number being multiplied.
• The exponent tells us how many times to multiply the base by itself.

In the expression \((-5)^2\),
- The base is \(-5\) and the exponent is 2. This means we multiply \(-5\) by itself once: \((-5) \times (-5)\).
Thus, understanding how exponents work helps clarify why certain expressions yield certain results. It's simply repeated multiplication.

Multiplying Negative Numbers

Negative numbers can be tricky, but there's a straightforward rule that helps simplify calculations: multiplying two negative numbers results in a positive number. This is because the negative signs "cancel out" each other. Consider the key points:

• \((-a) \times (-b) = ab\), meaning multiplying two negatives gives a positive product.
• Each negative switches the sign, so two flips return to positive.

Applying this to \((-5) \times (-5)\):
- The negative sign multiplied by another negative sign becomes positive.
- So, \((-5) \times (-5) = 25\). This rule always converts double negatives into positives.

Order of Operations

When faced with mathematical expressions, following the correct order of operations is crucial to arriving at the right answer. It's often remembered by the acronym PEMDAS, which stands for:

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

This structured order ensures consistency across all mathematical computations. For the expression \((-5)^2\):
- Since it involves an exponent, handle it before any addition or subtraction that might be involved in a larger expression.
- Thus we calculate the power first, as \((-5) \times (-5)\), followed by any other operations required if this was part of a more extensive equation.
Remember, working in the correct order prevents mistakes and ensures accuracy every time.