Problem 19 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: Write the multiplication

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

Step 3: Multiply two terms at a time

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

Step 4: Multiply the next two terms

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

Step 5: Multiply the results from Step 3 and Step 4

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

Step 6: Write the final answer

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

Key Concepts

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.