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Chapter 10: Problem 62

In the following exercises, simplify each expression with exponents. $$(-3)^{5}$$

Short Answer

Expert verified
-243

Step by step solution

01

Understand the Problem

The expression \((-3)^5\) involves a base of -3 raised to the power of 5. The goal is to simplify this expression.
02

Apply the Exponent

For an expression of the form \(a^n\), where \(a\) is the base and \(n\) is the exponent, you multiply the base by itself \(n\) times. Here, \(a = -3\) and \(n = 5\).
03

Calculate the Multiplications

Multiply -3 by itself 5 times: \[ (-3) \times (-3) \times (-3) \times (-3) \times (-3) \].
04

Simplify the Result

Calculate the multiplication: \[ (-3) \times (-3) = 9 \], \[ 9 \times (-3) = -27 \], \[ -27 \times (-3) = 81 \], \[ 81 \times (-3) = -243 \]. So, \((-3)^5 = -243\).

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Key Concepts

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

Understanding Exponents
When dealing with exponents, it's essential to understand what they represent.
An exponent indicates how many times a number, known as the base, is multiplied by itself.
For example, in the expression \((-3)^5\), -3 is the base, and 5 is the exponent.
This means that -3 is multiplied by itself five times.
This concept can be used to simplify expressions efficiently.
The Multiplication Process for Exponents
The key to simplifying expressions with exponents is breaking down the multiplication process.
When you have an exponent, you perform the multiplication step by step.
Let's follow the steps of our example \((-3)^5\):
  • First multiplication: (-3) × (-3) = 9
  • Second multiplication: 9 × (-3) = -27
  • Third multiplication: -27 × (-3) = 81
  • Fourth multiplication: 81 × (-3) = -243
As you can see, we multiply the result of each step by the base -3.
Doing it step-by-step helps avoid mistakes and ensures precision in the final result.
Base and Exponent Clarified
In any expression featuring exponents, differentiating between the base and the exponent is crucial for proper simplification.
The base is the number that is being multiplied multiple times.
The exponent tells us the number of times we multiply this base by itself.

For \((-3)^5\),
  • The base is -3.
  • The exponent is 5, meaning we'll multiply -3 by itself five times.
Always remember this rule: If the base is negative and the exponent is odd, the result will be negative.
If the base is negative and the exponent is even, the result will be positive.
By understanding these roles, you can simplify any similar expressions with confidence.

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