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Chapter 1: Problem 19

Simplify. $$ (-5)^{4} $$

Short Answer

Expert verified
The simplified form of \((-5)^{4}\) is 625.

Step by step solution

01

Understand the base and exponent

The expression \((-5)^{4}\) consists of a base of -5 and an exponent of 4 which means -5 will be multiplied by itself 4 times.
02

Write the multiplication

Write out the multiplication as follows: \((-5) \times (-5) \times (-5) \times (-5)\).
03

Multiply two terms at a time

First, multiply the first two terms: \( (-5) \times (-5) = 25 \).
04

Multiply the next two terms

Next, multiply the remaining terms: \( (-5) \times (-5) = 25 \).
05

Multiply the results from Step 3 and Step 4

Finally, multiply the two results: \( 25 \times 25 = 625 \).
06

Write the final answer

Thus, the simplified form of \((-5)^{4}\) is 625.

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Key Concepts

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

Simplifying Exponents
Exponents are a shorthand way to express repeated multiplication of the same number. In the expression \((-5)^{4}\), the 4 is the exponent and tells us how many times to multiply -5 by itself.
Breaking it down:
  • \((-5)^{4}\) means -5 will be multiplied by itself 4 times: \((-5) \times (-5) \times (-5) \times (-5)\).
  • It's helpful to simplify in steps to avoid mistakes.
Following these steps ensures accuracy and understanding.
Negative Bases
When working with negative bases, the position of parentheses is important.
Consider \((-5)^{4}\) vs \(-5^{4}\):
  • \((-5)^{4}\): The base is negative 5 and is raised to the power of 4.
  • \(-5^{4}\): The base is 5, and only the result is negative.
With \((-5)^{4}\), the negative sign is included in each multiplication:
First, multiply two -5s: \((-5) \times (-5) = 25\). Repeat for the other pair: \((-5) \times (-5) = 25\).
Finally, multiply the results: \(25 \times 25 = 625\).
Multiplication of Integers
Multiplication of integers follows specific rules when dealing with positive and negative numbers.
Key points to remember:
  • Multiplying two positive integers or two negative integers results in a positive integer.
  • Multiplying a positive integer with a negative integer results in a negative integer.
For example: \( (-5) \times (-5) = 25\) because both numbers are negative.
This same rule applies when multiplying further results:
\(25 \times 25 = 625\) results in a positive outcome.

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