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Chapter 5: Problem 19

Simplify each expression. Write each result using positive exponents only. $$ (-3)^{-2} $$

Short Answer

Expert verified
((-3)^{-2} = \frac{1}{9}\).

Step by step solution

01

Understand the Concept of Negative Exponents

A negative exponent indicates that we take the reciprocal of the base and then apply the positive exponent. For example, for any non-zero number \(a^{-n} = \frac{1}{a^n}\).
02

Apply the Reciprocal Rule

Given the expression \((-3)^{-2}\), apply the negative exponent rule. Convert the negative exponent by taking the reciprocal: \((-3)^{-2} = \frac{1}{(-3)^2}\).
03

Simplify the Expression

Calculate \((-3)^2\). The exponent 2 means we multiply the base \(-3\) by itself: \((-3) \times (-3) = 9\). Thus, \((-3)^2 = 9\).
04

Write the Final Answer

Substitute back into the fraction the simplified expression: \(\frac{1}{9}\). This is the simplified form using only positive exponents.

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Key Concepts

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

Understanding the Reciprocal Rule for Negative Exponents
Negative exponents can seem confusing at first, but they actually follow a simple rule: they signal a need to take the reciprocal of the base with a positive exponent. To break it down, any base raised to a negative exponent, like \(a^{-n}\), is equivalent to \(\frac{1}{a^n}\).

This means you will flip the base into the denominator and switch the exponent to a positive one.
  • "Reciprocal" here refers to swapping a number from the numerator to the denominator (or vice-versa) in a fraction.
  • For example, \((-3)^{-2}\) becomes \(\frac{1}{(-3)^2}\).
By easily transforming negative exponents into fractions, you simplify the numbers to work with clean positive exponents.
Exploring Positive Exponents
Positive exponents tell us to multiply a number by itself a specific number of times. In the expression \((-3)^2\), the positive exponent of 2 means you multiply the base \(-3\) two times: \((-3) \times (-3)\).

Multiplying these together results in a positive product because negative times negative equals positive. So,
  • \((-3)^2 = 9\).
Such calculations allow you to express numbers in their simplest forms. Always ensure to manage signs correctly, especially when negative bases are involved. Mistakes here usually lead to incorrect signs in your final answer.
Grasping Exponentiation Concepts
Exponentiation is a fundamental math operation another step up from multiplication. It's important to understand this operation, as exponents are used widely in algebra and beyond.

Here's a quick rundown:
  • An exponent indicates how many times to use the base in a multiplication.
  • For a positive integer exponent \(n\), the expression \(a^n\) means \(a\) multiplied by itself \(n\) times.
  • Negative exponents indicate taking the reciprocal of the base with its corresponding positive exponent, as explained earlier.
Mastering these basics opens the door to understanding more complex mathematical problems and enhances your problem-solving ability.

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Most popular questions from this chapter

Multiply. $$ 3 x(2 x) $$

Evaluate each expression using exponential rules. Write each result in standard form. $$ \frac{0.4 \times 10^{5}}{0.2 \times 10^{11}} $$

It was stated earlier that for an integer \(n\) \(x^{-n}=\frac{1}{x^{n}}, \quad x \neq 0\) Explain why \(x\) may not equal 0 .

Subtract \(y\) from \(\left(y^{2}-4 y+1\right)\)

Add or subtract as indicated. $$ \left(x^{2}+2 x y-y^{2}\right)+\left(5 x^{2}-4 x y+20 y^{2}\right) $$
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