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Chapter 1: Problem 23

Evaluate. $$(-6)^{2}$$

Short Answer

Expert verified
The square of -6 is 36, as \((-6)^{2} = (-6) \times (-6) = 36\).

Step by step solution

01

Understand the problem

We are given the expression \((-6)^{2}\), which means -6 multiplied by itself.
02

Apply the exponent rule

To find the square of -6 (\((-6)^{2}\)), we simply multiply -6 by itself: \[ (-6)^{2} = (-6) \times (-6) \]
03

Multiply the two negative numbers

When multiplying two negative numbers, the result is positive. So: \[ (-6) \times (-6) = 36 \]
04

State the final answer

The square of -6 is 36: \[ (-6)^{2} = 36 \]

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Key Concepts

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

Evaluating Expressions with Exponents
Evaluating expressions can often seem intimidating, but with a clear understanding of the process, it becomes quite manageable. Let's consider the given expression \((-6)^{2}\). Here, \(-6\) is our base, and \(^2\) is the exponent. The exponent tells us that the base should be multiplied by itself. Hence, \((-6)^{2}\) means \(-6 \times -6\). The key is to remember that the exponent indicates how many times to repeat the multiplication by the base.
  • Look at the base and the exponent. In this case, \(-6\) and \(^2\).
  • The exponent 2 means the base \(-6\) is used twice in multiplication.
Evaluating expressions correctly involves identifying these components and applying the exponent rules accurately.
Understanding Negative Numbers
Negative numbers might seem tricky at first, especially when combined with exponents. A negative number is simply a number less than zero. They behave differently than positive numbers during mathematical operations. In our example, \(-6\) is a negative number raised to an exponent. It's essential to note that the negative sign belongs with the base number. This impacts how we calculate the expression.
  • When a negative number is squared, its negative sign is part of the base.
  • Squaring a negative number results in a positive product because the negatives cancel each other out.
  • Recognize that \((-6)^{2}\) includes two negative multiplications.
Understanding these principles ensures you handle negative numbers confidently, especially with exponents.
Multiplication Rules and Negative Numbers
Multiplication rules are fundamental in solving problems with exponents, especially when dealing with negative numbers. One of the crucial rules is that multiplying two negative numbers results in a positive number. Let's see how this applies to \((-6)^{2}\).
  • First negative: \(-6\)
  • Second negative: again, \(-6\).
  • Multiply them: \(-6 \times -6 = 36\).
Why does it become positive? Because multiplying two negatives cancels out the signs, resulting in a positive product. Therefore, understanding multiplication rules is essential for correctly evaluating expressions involving negative numbers and exponents.

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