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Chapter 1: Problem 37

Evaluate each expression. See Example 6 $$ (-6)^{2} $$

Short Answer

Expert verified
The value of \((-6)^2\) is 36.

Step by step solution

01

Identify the Base and the Exponent

In the given expression \((-6)^2\), the base is \(-6\) and the exponent is \(2\). The exponent tells us how many times to multiply the base by itself.
02

Multiply the Base by Itself

Since the exponent is \(2\), we multiply \(-6\) by itself: \((-6) \times (-6)\).
03

Calculate the Product

Calculate the multiplication: \((-6) \times (-6) = 36\).Multiplying two negative numbers results in a positive number.

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Key Concepts

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

Understanding Base and Exponent
When working with exponentiation, it is important to understand the terms "base" and "exponent." The base is the number that is being multiplied and the exponent indicates how many times the base is used as a factor. In the expression \[ (-6)^2 \], we have:
  • Base: The base is \(-6\). This is the number that will be multiplied by itself a certain number of times, determined by the exponent.

  • Exponent: The exponent is \(2\). It tells us that the base \(-6\) will be multiplied by itself two times.
Since the exponent is \(2\), the operation we need to perform is simply: \[ (-6) \times (-6). \]Mastering base and exponent concepts allows us to solve many mathematical problems involving powers, thus making calculations faster and efficient.
Concept of Multiplication of Integers
Multiplication is one of the fundamental operations in mathematics. When we talk about the multiplication of integers, we mean taking one integer and adding it to itself multiple times, as dictated by another integer. For example, in the expression:\[ (-6) \times (-6), \] we multiply the integer \(-6\) by itself because the exponent indicates the number of repetitions. It's helpful to think of multiplication as repeated addition, though when dealing with negative numbers, this becomes a bit more involved. Here's how it works in practice:
  • Two positive numbers multiply to give a positive product.

  • Two negative numbers also multiply to give a positive product.

  • But, if one number is negative and the other is positive, the product is negative.
Understanding these rules is essential for correctly performing multiplication tasks involving integers.
Working with Negative Numbers in Multiplication
Negative numbers add a twist to multiplication. They have their own set of rules to follow during multiplication. In the expression\((-6) \times (-6)\), both numbers are negative.When multiplying negative numbers:
  • Negative Times Negative: Always results in a positive number. This is because the negatives "cancel each other out." So \((-6) \times (-6) = 36\).

  • Negative Times Positive or Vice Versa: Results in a negative number. However, this isn't relevant when the base is negative and the exponent is even, as both negatives will cancel.
By learning how to handle negative numbers during multiplication, we can easily evaluate expressions, no matter how confusing they might initially seem. The key takeaway here is that a negative times a negative becomes a positive, which is crucial for correctly solving such problems.

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Most popular questions from this chapter

Write each of the following as an algebraic expression. See Examples 12 and 13. Two numbers have a sum of \(25 .\) If one number is \(x,\) represent the other number as an expression in \(x\).

Complete the statement to illustrate the given property. \(8+0=\quad\)______ Additive identity property

Use a calculator to approximate each square root. simplify the expression. Round answers to four decimal places. \(\sqrt{273}\)

In your own words, explain why every set is a subset of itself.

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ -4(y z+3)-7 y z+1+y^{2} $$
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