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Chapter 1: Problem 6

Evaluate each exponential expression. $$(-10)^{2}$$

Short Answer

Expert verified
The result of \( (-10)^{2} \) is 100.

Step by step solution

01

Understand the exponent

An exponent refers to the number of times a number is multiplied by itself. For example, in this case, \( (-10)^{2} \) means \(-10\) is multiplied by itself once.
02

Use the property of the exponent

In general, a negative number raised to an even power equals a positive number. Thus, apply this definition to this problem: \( (-10)^{2} = -10 * -10 \).
03

Calculate the multiplication

Calculate the multiplication: -10 * -10 = 100.

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Key Concepts

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

Negative Numbers
Negative numbers are less than zero and are typically represented with a minus sign. They are found to the left of zero on the number line. When we perform operations on them, certain rules need to be applied. It's essential to understand these rules because they impact how calculations are done.

  • When adding a negative number, it is similar to subtracting its absolute value.
  • Subtracting a negative number is like adding its absolute value.
  • Multiplying two negative numbers results in a positive number.
  • Dividing two negative numbers also results in a positive number.
In the context of exponents, negative numbers can have interesting properties, especially when paired with even or odd exponents. Understanding these rules lays a foundation for correctly evaluating expressions involving negative numbers.
Multiplication
Multiplication is an arithmetic operation that combines groups of equal size. It can be thought of as repeated addition. For example, 4 times 3 (written as \(4 \times 3\)) means adding four three times: \(4 + 4 + 4 = 12\). When working with negative numbers, multiplication follows specific rules that help determine the sign of the product.

  • A positive number times a positive number equals a positive number.
  • A positive number times a negative number equals a negative number.
  • A negative number times a negative number equals a positive number.
In the given exercise, we multiply \(-10\) by itself. Because both factors are negative and the rules of multiplication state that a negative times a negative is positive, the answer becomes \(+100\). Consistent practice with these rules will build proficiency in calculating expressions correctly.
Even Powers
Even powers refer to exponents that are even numbers. An exponent represents how many times to use the base number in a multiplication. If you apply an even exponent to a number, the base is multiplied by itself an even number of times, such as 2, 4, 6, etc.

When a negative number is raised to an even power, the result is always positive. This is because:
  • Multiplying a negative number by itself an even number of times results in a positive product.
  • Every pair of negative numbers multiplied yields a positive result.
For example, with \((-10)^{2}\), the even power 2 tells us to multiply \(-10\) by itself: \(-10 \times -10 = 100\). This concept is crucial for algebra and ensures you're calculating expressions with negative bases and even exponents correctly.

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Most popular questions from this chapter

Write each phrase as an algebraic expression. a loss of half of an investment of \(d\) dollars

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The product of \(-6\) and \(-4,\) decreased by \(-5\)

Use a calculator to find a decimal approximation for each irrational number, correct to three decimal places. Between which two integers should you graph each of these numbers on the number line? $$\sqrt{3}$$

Describe how the inverse property of addition $$a+(-a)=0$$ can be shown on a number line.

Exercises \(150-152\) will help you prepare for the material covered in the next section. In each exercise, an expression with an exponent is written as a repeated multiplication. Find this product, indicated by a question mark. $$(-6)^{2}=(-6)(-6)=?$$
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