Problem 14 - Algebra 1

Question

Evaluate the exponential expression. Write fractions in simplest form. $$4^{-2}$$

Step-by-Step Solution

Answer

The simplest form of \(4^{-2}\) is \(\dfrac{1}{16}\)

Step 1: Understand Exponential Expression

The given expression is \(4^{-2}\). In an exponential expression, the number below the exponent (in this case 4) is called the base while the number on top (in this case -2) is called the exponent. The negative sign in the exponent tells that the base needs to be taken reciprocal.

Step 2: Inverse base with negative exponent

The negative sign in the -2 exponent tells us that we need to take the reciprocal of the base. So, \(4^{-2}\) is equivalent to \(\dfrac{1}{4^2}\).

Step 3: Calculate

Now, proceed to calculate the expression by squaring the 4 in the denominator. This gives us \(\dfrac{1}{4^2} = \dfrac{1}{16}\).

Key Concepts

Negative Exponents

Negative exponents might seem intimidating at first, but they have a simple rule. Whenever you see a negative exponent, it indicates that you need to take the reciprocal of the base raised to the corresponding positive exponent.
For example, in the expression \(4^{-2}\), the "-2" tells us to take the base of 4 and use its reciprocal instead of raising it.

• Negative exponent: Switch places in a fraction.
• Turn into a positive exponent by "flipping" the fraction.

Understanding this can make complex expressions much more manageable and help simplify the expressions quickly.

Reciprocal

A reciprocal is simply a way to 'flip' a fraction. For a number \(a\), its reciprocal is \(\frac{1}{a}\). When applied to whole numbers like 4, the reciprocal is \(\frac{1}{4}\).
This concept is crucial when dealing with negative exponents, as they require us to work with reciprocals. In our example, \(4^{-2}\) becomes \(\frac{1}{4^2}\).

• Transforms multiplication into division with fractions.
• Used frequently in division of numbers and algebraic expressions.

Using reciprocals allows us to convert negative exponents into easier positive exponents, ensuring straightforward calculations.

Simplest Form

Simplifying an expression to its simplest form means making the expression as compact as possible without changing its value.
This often involves combining like terms or, in the case of fractions, reducing them to their simplest terms.

• For the fraction \(\frac{1}{16}\), no further simplification can occur because it's already in its simplest form.
• Simplest form helps in clearly understanding and comparing different fractions or terms.

Reaching the simplest form is often the final step in evaluating expressions, providing a concise, clear result.

Fractions

Fractions represent a part of a whole and consist of a numerator and a denominator. They are a way to express division of integers. When dealing with negative exponents, we often end up with fractions as part of the solution.
For instance, with \(4^{-2}\), we got \(\frac{1}{16}\). Here, 1 is the numerator and 16 is the denominator.

• Fraction \(\frac{1}{16}\) represents one part of sixteen equal parts.
• It's crucial to understand fractions to work efficiently with algebraic expressions and solve problems.

Being comfortable with fractions can make the process of simplifying exponential expressions much easier and allow for handling more complex mathematical problems.