Problem 21 - Algebra and Trigonometry

Question

Evaluate each expression. $$ 2^{-2}+2^{-3} $$

Step-by-Step Solution

Answer

\(\frac{3}{8}\)

Step 1: Understand the negative exponent rule

When we encounter a negative exponent, like \(2^{-n}\), it means we take the reciprocal of the base raised to the positive exponent. Specifically, \(2^{-n} = \frac{1}{2^n}\).

Step 2: Convert negative exponents to fractions

Apply the negative exponent rule to the given expression: \(2^{-2}\) becomes \(\frac{1}{2^2}\) and \(2^{-3}\) becomes \(\frac{1}{2^3}\).

Step 3: Calculate the powers of 2

Calculate \(2^2\) and \(2^3\). We have \(2^2 = 4\) and \(2^3 = 8\).

Step 4: Simplify fractions

Convert the expressions by replacing the powers of 2 with their calculated values: \(\frac{1}{2^2}\) becomes \(\frac{1}{4}\) and \(\frac{1}{2^3}\) becomes \(\frac{1}{8}\).

Step 5: Add the fractions

To add \(\frac{1}{4}\) and \(\frac{1}{8}\), find a common denominator. The common denominator is 8. Convert \(\frac{1}{4}\) to \(\frac{2}{8}\). The sum becomes \(\frac{2}{8} + \frac{1}{8} = \frac{3}{8}\).

Step 6: Final result

The simplified result of the expression \(2^{-2} + 2^{-3}\) is \(\frac{3}{8}\).

Key Concepts

Fraction Addition

Adding fractions involves combining ratios with different denominators. To add fractions, find a common denominator, which is the least common multiple of the denominators.
This ensures you are working with equivalent fractions.Steps to add fractions:

• Identify the denominators of the fractions you want to add.
• Find the least common multiple (LCM) of these denominators to use as a common denominator.
• Rewrite each fraction as an equivalent fraction with this common denominator.
• Now that the fractions share a common denominator, add the numerators while keeping the denominator the same.
• Simplify the resulting fraction if possible.

In the example, \(\frac{1}{4} + \frac{1}{8}\), the LCM of 4 and 8 is 8. Convert \(\frac{1}{4}\) to \(\frac{2}{8}\) because \(1 \times 2 = 2\) and \(4 \times 2 = 8\). Now, adding \(\frac{2}{8}\) and \(\frac{1}{8}\) gives \(\frac{3}{8}\).

Powers of 2

The powers of 2 are results of multiplying the number 2 by itself a certain number of times. When we talk about a power of 2, we use exponents to express how many times 2 is used as a factor.
For any positive integer n, \(2^n\) refers to multiplying 2 by itself n times.Key powers of 2 used frequently:

• \(2^0 = 1\)
• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)

Remembering these basic powers can speed up calculations and make simplification easier. In this exercise, calculating \(2^2 = 4\) and \(2^3 = 8\) helped in finding the fractional values \(\frac{1}{4}\) and \(\frac{1}{8}\).

Exponent Rules

Exponent rules, particularly with negative exponents, guide us on how to manipulate expressions involving exponents. Negative exponents indicate a reciprocal action.
Here are some essential exponent rules:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^n = a^n \times b^n\)
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)

The rule for a negative exponent \(2^{-n} = \frac{1}{2^n}\) transforms a power into a fraction, effectively rewriting \(2^{-2}\) as \(\frac{1}{2^2}\) and \(2^{-3}\) as \(\frac{1}{2^3}\). Understanding and applying this rule allows for the simplification and evaluation of expressions with ease.