Problem 8 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ 4^{-1}+4^{-2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5}{16}\).

Step 1: Identify the Problem Statement

We need to simplify the expression \(4^{-1} + 4^{-2}\) and express the result using positive exponents only.

Step 2: Rewrite Negative Exponents

Convert each negative exponent into a positive exponent using the property \(a^{-n} = \frac{1}{a^n}\). This means:\[4^{-1} = \frac{1}{4^1} = \frac{1}{4}\]\[4^{-2} = \frac{1}{4^2} = \frac{1}{16}\]

Step 3: Find a Common Denominator

The fractions \(\frac{1}{4}\) and \(\frac{1}{16}\) have different denominators. To add them, find a common denominator. Here, 16 is the least common denominator:\[\frac{1}{4} = \frac{4}{16}\]

Step 4: Add the Fractions

Now that both fractions have the same denominator, add them:\[\frac{4}{16} + \frac{1}{16} = \frac{4 + 1}{16} = \frac{5}{16}\]

Step 5: Simplify the Result

The expression \(\frac{5}{16}\) is already simplified and is in its lowest terms, written using positive exponents.

Key Concepts

Negative Exponents

Negative exponents often seem intimidating, but they are quite manageable once you understand the concept. When you see a base raised to a negative exponent, like in the expression \(4^{-1}\), you can convert it into a fraction. This is achieved by taking the reciprocal of the base and changing the exponent to positive.

For instance, \(4^{-1}\) becomes \(\frac{1}{4^1}\), which is simply \(\frac{1}{4}\). Similarly, \(4^{-2}\) turns into \(\frac{1}{4^2}\), which calculates to \(\frac{1}{16}\).

Here's a quick summary to help you remember:

• For any number \(a\), \(a^{-n} = \frac{1}{a^n}\).
• Changing a negative exponent to a positive one involves transforming it into a fraction.

Negative exponents are a method of expressing division in a compact form.

Adding Fractions

Adding fractions involves a few simple steps, but it requires that the fractions have the same denominator. Here's why: the denominator signifies the total number of parts a whole is divided into, and for two or more fractions to be combined, the fragments must be of uniform size.

Once all the fractions involved share a common denominator, adding them becomes straightforward. You simply sum their numerators, while the common denominator remains unchanged. For example, once you have \(\frac{4}{16}\) and \(\frac{1}{16}\), you can proceed to add them:

• The new numerator becomes \(4 + 1\).
• The denominator stays 16, making the sum \(\frac{5}{16}\).

Remember, the sum of fractions is only as simple as ensuring their pieces match.

Common Denominators

Finding a common denominator is a crucial step when adding fractions with different denominators. A common denominator is a shared multiple of the denominators of given fractions. By converting fractions to have this common denominator, you are creating an equivalent scenario where the fractions have parts of equal size.

In our exercise, the fractions \(\frac{1}{4}\) and \(\frac{1}{16}\) need a common denominator. The least common multiple (LCM) of 4 and 16 is 16. You adjust \(\frac{1}{4}\) by expanding it:

• Multiply the numerator and the denominator of \(\frac{1}{4}\) by 4 to get \(\frac{4}{16}\).

Now both fractions \(\frac{4}{16}\) and \(\frac{1}{16}\) can be easily added, as they share the same denominator.
Understanding common denominators and how to find them is essential for simplifying expressions that involve fraction addition.