Question

Evaluate the expression. Write fractional answers in simplest form.\(4^{-1}-2^{-2}\)

Short answer

The answer is 0.

Step-by-step solution

Understand the Properties of Negative Exponents

In mathematics, a negative exponent represents a fraction, where the base of the exponent is in the denominator. So, \(4^{-1}\) is equivalent to \(\frac {1}{4}\), and \(2^{-2}\) is equivalent to \(\frac {1}{4^2}\) or \(\frac {1}{4}\).

Substitution

Substitute \(4^{-1}\) and \(2^{-2}\) with their respective fractional equivalents. The expression becomes \(\frac {1}{4} - \frac {1}{4}\).

Perform the Subtraction

Subtract the fractions. The denominator for both fractions is the same, so subtract the numerators and place the result over the common denominator. The answer becomes \(\frac {1-1}{4} = \frac {0}{4}\).

Simplify

Any number multiplied by zero results in zero. So, the fraction \(\frac {0}{4}\) simplifies to zero.

Key concepts

Negative exponents

When it comes to negative exponents, the key to understanding them lies in recognizing their relationship to fractions. A negative exponent tells us to take the reciprocal of the base and then apply the positive exponent. For example, an expression like \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\), where \(a\) is the base and \(n\) is the positive exponent. Taking the example from the exercise, \(4^{-1}\) can be rewritten as \(\frac{1}{4}\). This approach transforms an expression with a negative exponent into a fraction, making it markedly easier to manage as you perform arithmetic operations like addition or subtraction.

Simplifying fractions

When we simplify fractions, we're looking to express them in the most basic form possible, where the numerator and denominator have no common factors other than 1. It's important to pay attention to zero because any fraction with 0 as its numerator is actually equal to zero, no matter what the denominator is. This is particularly relevant when evaluating expressions like in our exercise, where subtraction may result in a numerator of zero, rendering the entire fraction zero.

Fractional notation

The fractional notation is a way to represent division between two numbers, a numerator and a denominator. Understanding this concept is particularly useful when dealing with negative exponents, as they often result in fractional expressions. In our exercise, for instance, recognizing that \(2^{-2}\) is the same as \(\frac{1}{2^2}\) helps in converting the expression into a form that is easier to work with. The key in fractional notation is to maintain clarity and consistency when performing operations like simplification or comparing fractions.

Algebraic subtraction

Algebraic subtraction involving fractions requires a common denominator to perform the operation directly on numerators. In our initial expression \(4^{-1}-2^{-2}\), we simplify the negative exponents to get \(\frac{1}{4}-\frac{1}{16}\). To subtract these fractions, one would need to find a common denominator. In this case, the common denominator is 16, and the expression would be rewritten as \(\frac{4}{16}-\frac{1}{16}\). This results in \(\frac{3}{16}\) after performing the subtraction of the numerators. It's essential to be precise with this process to ensure the final answer is in its simplest form.