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Chapter 1: Problem 36

Evaluate each expression. See Example 6 $$ (-7)^{2} $$

Short Answer

Expert verified
49

Step by step solution

01

Understand the Expression

We are given the expression \((-7)^2\). This means we need to square the number -7, which involves multiplying the number by itself.
02

Identify the Base and the Exponent

In the expression \((-7)^2\), -7 is the base, and 2 is the exponent. The exponent tells us how many times to use the base in the multiplication.
03

Square the Base

To square the base \(-7\), we multiply \(-7\) by itself: \(-7 \times -7\).
04

Calculate the Multiplication

Calculate the product: \(-7 \times -7 = 49\). The multiplication of two negative numbers results in a positive number.
05

Finalize the Answer

The result of squaring \(-7\) is 49. Therefore, \((-7)^2 = 49\).

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Key Concepts

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

Squaring Numbers
Squaring a number is a way to find its value when multiplied by itself. For example, squaring the number 3 involves calculating \(3 \times 3\). The result, which is 9, is the square of 3. Similarly, for any number \(x\), its square is denoted by \(x^2\) and is equal to \(x \times x\). This concept is important because it shows how squaring can change the magnitude of a number, sometimes leading to much larger values.
  • This operation is very common in mathematics, especially in algebra and geometry.
  • Squaring applies to both positive and negative numbers.
  • It's a foundational concept that will be built upon in more advanced topics like quadratic equations.
Understanding squaring helps you tackle various math problems more effectively, prepare for more complex topics, and enhance your number sense skills.
Negative Numbers
Negative numbers are numbers less than zero and are written with a minus sign (鈭�) in front of them, like \(-1, -2, \) or \(-7\). These numbers work a bit differently than positive numbers, especially when it comes to operations like multiplication and squaring.
  • When you square a negative number, the two negative signs cancel each other, resulting in a positive number. For example, squaring \(-7\) gives \(49\) because \((-7) \times (-7) = 49\).
  • Understanding negative numbers is important for solving equations and working with real-life situations like debts or below-zero temperatures.
In practical terms, working with negative numbers can sometimes feel tricky because we are often used to dealing only with positive values. However, mastering these concepts can greatly improve your mathematical ability and problem-solving skills.
Multiplying Integers
Multiplying integers involves several basic rules, especially when incorporating negative numbers. When multiplying two integers:
  • A positive times a positive yields a positive result.
  • A negative times a negative also produces a positive result, as seen with \((-7) \times (-7) = 49\).
  • A positive times a negative, or vice versa, yields a negative result. For example, \(5 \times (-2) = -10\).
These rules help clarify why squaring a negative number results in a positive product. Understanding these fundamental operations is crucial for tackling more complex algebraic expressions and equations.

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Most popular questions from this chapter

Choose the fraction (s) equivalent to the given fraction. (There may be more than one correct choice.) \(\frac{-x}{y} \quad\) a. \(\frac{x}{-y}\) b. \(-\frac{x}{y} \quad\) c. \(\frac{x}{y}\) d. \(\frac{-x}{-y}\)

Choose the fraction (s) equivalent to the given fraction. (There may be more than one correct choice.) $$ \begin{aligned} &\frac{5}{-(x+y)} \quad \text { a. } \frac{5}{(x+y)} \quad \text { b. } \frac{-5}{(x+y)}\\\ &\frac{-1}{-(x+y)} \cdot \frac{-5}{(x+y)} \end{aligned} $$

Evaluate each expression when \(x=9\) and \(y=-2 .\) See Example 12 $$ 9 x-6 y $$

Translating Write each phrase as an algebraic expression. Use the variable \(x\) to represent each unknown number. See Example 8 . The quotient of five and the difference of four and a number.

Simplify each expression. See Examples I through 11 . $$ \frac{\frac{1}{3} \cdot 9-7}{3+\frac{1}{2} \cdot 4} $$
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