Problem 103 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression.$$ 4^{-1} $$

Step-by-Step Solution

Answer

Answer: $$\frac{1}{4}$$.

Step 1: Understanding Negative Exponents

In this expression, we have a negative exponent (-1). A negative exponent means that we need to take the reciprocal (or the inverse) of the base number. So, $$4^{-1} = \frac{1}{4^1}$$.

Step 2: Evaluating the Expression

Now that we have our expression written as a fraction, we can simplify it further. As we know, $$4^1 = 4$$. Therefore, the expression simplifies to $$\frac{1}{4}$$.

Key Concepts

Understanding Negative Exponents

When you come across an expression with a negative exponent, such as \(4^{-1}\), it's crucial to grasp what this notation actually signifies. A negative exponent instructs you to do the opposite of multiplying the base number by itself; instead, you must take the reciprocal of the base number and raise it to the opposite positive exponent. To put it simply, having a negative exponent means you divide 1 by the base number raised to the corresponding positive exponent.

Let's demystify the process of evaluating negative exponents with a clear example: For \(4^{-1}\), you first identify the base number (which is 4) and then take the reciprocal of that base. This gives you \(1/4\), and because the exponent is -1, you don't have to multiply it by itself at all. Thus, \(4^{-1} = 1/4\), which is the same as saying 'one divided by four.' This concept is a fundamental part of algebra, and understanding it can vastly simplify complex expressions.

The Reciprocal of Base Number

A reciprocal is simply a way of expressing one number divided by another number. In the context of negative exponents, when you take the reciprocal of a base number, you are essentially 'flipping' the fraction. The reciprocal of a whole number like 4 is \(1/4\), and for a fraction like \(a/b\), the reciprocal would be \(b/a\). It is also worth noting that the product of a number and its reciprocal is always 1.

For instance, in evaluating \(4^{-1}\), the first step is to transform the number 4 into its reciprocal form, 1/4. When dealing with more complex numbers, always remember that the reciprocal involves switching the numerator and the denominator if the base is already a fraction. This strategy is particularly important when simplifying expressions as it sets the stage for further manipulation of the numbers involved.

Simplifying Expressions with Negative Exponents

Simplifying expressions is an essential skill in algebra, and it becomes even more crucial when dealing with negative exponents. The goal here is to create a more straightforward form of the expression that is easier to understand and evaluate. The process usually involves converting negative exponents into positive exponents by finding the reciprocal of the base number, as we've seen with \(4^{-1}\).

However, the process doesn't stop there. There can be additional steps depending on the complexity of the expression. For example, if you encounter an expression like \((2/3)^{-2}\), you would first find the reciprocal, which gives \((3/2)^{2}\), and then proceed to square both the numerator and the denominator. Simplifying algebraic expressions aids in both solving equations and understanding the relationships between variables. By practicing these simplifications, you develop a strong foundation that will help tackle more advanced mathematical concepts.