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Chapter 1: Problem 47

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

Short Answer

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

Step by step solution

01

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.
02

Multiply the Base by Itself

To evaluate \((-5)^2\), multiply the base \(-5\) by itself:\[-5 \times -5\].
03

Calculate the Product

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Evaluate each expression when \(x=-5, y=4,\) and \(t=10 .\) See Example 6. $$ x-y $$

In your own words, explain how to add two fractions with different denominators.

In \(2006,\) Gap Inc operated a total of 3054 stores worldwide. The following chart shows the store breakdown by brand. $$ \begin{array}{|l|l|} \hline \text { Brand } & {\text { Number of Stores }} \\ \hline \text { Gap(\mathrm{ Domestic) } } & {1335} \\ \hline \text { Gap (International) } & {256} \\ \hline \text { Banana Republic } & {498} \\ \hline \text { Old Navy } & {960} \\ \hline \text { Forth \& Towne } & {5} \\ \hline \text { Total } & {3054} \\ \hline \end{array} $$ What fraction of Gap-brand stores were Old Navy stores? Simplify this fraction.

Add or subcract as indicared. Write the answer in lower terms. See Erample 5 $$ \frac{17}{21}-\frac{10}{21} $$

Perform the following operations. Write answers in lowest terms. $$ \frac{2}{7}+\frac{4}{7} $$
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