Problem 11 - Introductory Algebra for College Students

Question

In Exercises \(1-14\), evaluate each exponential expression. $$-5^{4}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the meaning of \(a^{n}\)

When we have \(a^{n}\), it means that we are multiplying \(a\), \(n\) times. For example, if we have \(2^{3}\), it means that we multiply 2, three times, which is \(2*2*2\) equalling 8.

Step 2: Apply the rule to \(-5^{4}\)

According to the rule mentioned above, we multiply \(-5\), four times, which is \(-5*-5*-5*-5\).

Step 3: Perform multiplication

\(-5*-5 = 25, 25*-5 = -125, -125*-5 = 625\)

Key Concepts

Understanding Exponents

Exponents can be a tricky concept at first, but with some simple steps, they become easier to understand. An exponent is a small number written above and to the right of a base number. The exponent tells us how many times to multiply the base number by itself.
For example, when we look at an expression like \(2^3\), it means we multiply 2 by itself a total of three times. So, \(2 * 2 * 2\) equals 8.
It's like a special math shorthand that lets us quickly express large numbers.

• The base is the number being multiplied.
• The exponent is the small number indicating how many times the base is used as a multiplier.

Using exponents efficiently is crucial in simplifying expressions and solving algebraic problems.

Handling Negative Bases

When dealing with negative bases in exponential expressions, things get a bit more interesting. A negative base raised to an exponent means we multiply the negative number by itself for as many times as specified by the exponent. However, the outcome depends significantly on whether the exponent is even or odd.
When a negative base is raised to an even exponent, the result is positive. This happens because multiplying two negative numbers results in a positive number. Therefore, when you have an expression like \((-5)^4\), the result is positive because \((-5) * (-5) * (-5) * (-5) = 625\).

• If the exponent is odd, the result remains negative.
• If even, the result turns positive.

This is an essential concept, especially in solving algebraic equations where the base numbers might not always be positive.

The Role of Multiplication in Exponents

Multiplication plays a central role when we evaluate exponential expressions. Essentially, an exponent is a shorthand way to indicate how many times multiplication should occur.
Consider the situation where we've got the expression \(-5^4\), without proper parentheses, it actually means \(-(5 * 5 * 5 * 5)\).
Performing the multiplication steps one at a time, we have:

• \(-5 * -5 = 25\)
• \(25 * -5 = -125\)
• \(-125 * -5 = 625\)

This step-by-step multiplication process is crucial because exponents can grow larger very quickly.
Systematically breaking down each multiplication helps ensure accurate results, which is critical in both simple and complex calculations.