Problem 74 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (-5)^{2}(-5)^{-1} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((-5)^2(-5)^{-1}\). Answer: -5

Step 1: Rewrite the negative exponent as a positive exponent in a fraction

Recall that \(a^{-n} = \frac{1}{a^n}\). In our expression, we have (-5)^{-1}, which we can rewrite as: $$ (-5)^2(-5)^{-1} = (-5)^2\frac{1}{(-5)^1} $$

Step 2: Use the exponent rules to simplify numbers

Now we can simplify the numbers raised to their exponents. In this case, we have (-5)^2 which is equal to 25 and (-5)^1 which is equal to -5. So our expression becomes: $$ 25\frac{1}{-5} $$

Step 3: Multiply to get the final answer

Finally, we multiply 25 by the fraction: $$ 25\frac{1}{-5} = \frac{25}{-5} $$ Since 25 divided by -5 is equal to -5, our simplified expression is: $$ \boxed{-5} $$

Key Concepts

Negative Exponents

When you come across a negative exponent, it might seem a bit intimidating at first, but it's actually quite simple. A negative exponent indicates that instead of multiplying the base number by itself, you need to divide by that number. Specifically, a negative exponent means you take the reciprocal of the base raised to the positive of that exponent.
For example, in the expression \((-5)^{-1}\), the base is -5 and the exponent is -1. To rewrite this with a positive exponent, you use the rule:

• \(a^{-n} = \frac{1}{a^n}\)

The expression \((-5)^{-1}\) changes to \(\frac{1}{(-5)^1}\), simplifying the negative exponent into a fraction where the base is in the denominator. This transformation makes it easier to work with during calculations.

Applying Exponent Rules

Exponent rules are immensely helpful in simplifying expressions by providing clear methods to handle powers. One of the most used exponent rules is combining powers with the same base:

• \(a^m \times a^n = a^{m+n}\)

In our exercise, we have \((-5)^2\) and \((-5)^{-1}\). While these terms cannot be combined directly due to the negative exponent, we rewrite the negative exponent as discussed before: \((-5)^{-1} = \frac{1}{(-5)^1}\). Now, using our previous knowledge, we see \((-5)^2 = 25\) and using \((-5)^1 = -5\), simplifying further allows us to engage with these powers effectively.
Always remember the core exponent rules and understand:

• Add exponents when multiplying with the same base.
• Subtract exponents when dividing with the same base.

These rules provide a solid foundation for simplifying any expression regardless of complexity.

Simplifying Expressions

Once you've dealt with any negative exponents and applied your exponent rules, the next step is to carry out any necessary arithmetic to simplify the expression further. Let’s take our original expression and see how we can simplify it to its bare bones.
The first expression we rewrite is \((-5)^2(-5)^{-1}\), which transforms step-by-step into \(25 \cdot \frac{1}{-5}\), as shown before. This expression simplifies by multiplying 25 by the reciprocal of -5, resulting in:

• \(\frac{25}{-5} = -5\)

Simplifying expressions involves understanding both the numerical operations and the subtle rules surrounding powers. Completing these steps ensures your expression is as simplified as possible, reducing potential errors and providing clarity in the results.