Problem 47 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of \((-5)^2\) is 25.

Step 1: Identify the Base and the Exponent

In the expression \[(-5)^{2}\]the base is \(-5\) and the exponent is \(2\). This means you need to multiply \(-5\) by itself.

Step 2: Multiply the Base by Itself

To evaluate \((-5)^2\), multiply the base \(-5\) by itself:\[-5 \times -5\].

Step 3: Calculate the Product

Multiply the numbers:\(-5 \times -5 = 25\). Because multiplying two negative numbers results in a positive number, the answer is \(25\).

Key Concepts

Understanding Negative Numbers

Negative numbers might seem tricky at first, but they are just as simple to grasp once you get the hang of them. A negative number is any number that is less than zero and is represented by a minus sign, \(-\). This sign indicates that the number is below zero on the number line.
For example, in our exercise, the number -5 is negative. When we multiply or divide negative numbers, there are some simple rules to remember:

• Multiplying or dividing two positive numbers gives a positive result.
• Multiplying or dividing a positive number by a negative number results in a negative number.
• Multiplying or dividing two negative numbers results in a positive number.

Understanding these rules will help us solve exponent problems that involve negative numbers more easily.

Exploring Base and Exponent

In expressions with exponents, such as \((-5)^2\), it's essential to clearly identify both the base and the exponent. The base is the number being repeatedly multiplied, while the exponent indicates how many times the base is used in the multiplication.

In \((-5)^2\):

• Base: \(-5\), which means this is the number we will repeatedly multiply.
• Exponent: \2\, which tells us to multiply the base by itself once (since 2 - 1 = 1 additional multiplier after the initial number).

So, \((-5)^2\) tells us we need to calculate \(-5 \times -5\).
This understanding is fundamental as it allows us to break down the process and makes calculations more manageable.

The Magic of Multiplying Integers

Multiplying integers, especially when they're negative, follows straightforward rules, as previously mentioned. Let's apply these rules with a closer look at \((-5) \times (-5)\).
When multiplying two negative numbers, the product is positive. Visually, negative times negative flips the number to the positive side. Thus, \(-5 \times -5 = 25\).

Here's why:

• The first \(-5\) flips its direction from positive (5), pointing to the left on the number line.
• The second \(-5\) reverses it again, pointing back to the right, which returns it to positive.

So, when you multiply two negative integers, you get a positive result because two reversals bring you back to the original "direction." This clarity aids in solving many algebraic computations and ensures correct outcomes every time.