Problem 27 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ 2^{-2}+2^{-3} $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{3}{8}\).

Step 1: Understand Negative Exponents

Recall that a negative exponent indicates that you should take the reciprocal of the base raised to the corresponding positive exponent. For any base \(a\), \(a^{-n} = \frac{1}{a^n}\).

Step 2: Evaluate Each Term Separately

Evaluate \(2^{-2}\) and \(2^{-3}\) separately by converting them to their reciprocal forms:\[2^{-2} = \frac{1}{2^2} = \frac{1}{4}\]\[2^{-3} = \frac{1}{2^3} = \frac{1}{8}\]

Step 3: Add the Fractions

Add the fractions \(\frac{1}{4}\) and \(\frac{1}{8}\). Find the common denominator, which is 8:\[\frac{1}{4} = \frac{2}{8}\]Now add the fractions:\[\frac{2}{8} + \frac{1}{8} = \frac{3}{8}\]

Step 4: Verify the Result

Double-check that all steps have been correctly followed and that the common denominator was applied accurately. The sum of \(\frac{1}{4} = \frac{2}{8}\) and \(\frac{1}{8}\) confirms the result \(\frac{3}{8}\).

Key Concepts

Reciprocal Fractions

When it comes to negative exponents, the concept of reciprocal fractions plays a key role in simplifying expressions. A reciprocal is essentially flipping a fraction, and it is important to understand how this ties into converting negative exponents. Consider a base number with a negative exponent, such as \(a^{-n}\). This can be rewritten as a reciprocal of the base raised to a positive exponent: \(\frac{1}{a^n}\).
This means that for any negative exponent, find its reciprocal by taking one over the base raised to the positive version of that exponent. For example, the expression \(2^{-2}\) becomes \(\frac{1}{2^2}\), or \(\frac{1}{4}\). Likewise, \(2^{-3}\) converts to \(\frac{1}{2^3}\), which equals \(\frac{1}{8}\).
Understanding this conversion is crucial for evaluating and simplifying expressions involving negative exponents.

Fraction Addition

Adding fractions may seem tricky at first, but with a basic understanding of the procedure, it becomes straightforward. The key is to find a common denominator — a step that simplifies the addition process. Take for instance the fractions \(\frac{1}{4}\) and \(\frac{1}{8}\). To add these, the denominators must be the same.
To do this, find the least common multiple of the denominators. For \(4\) and \(8\), the smallest number they both divide into evenly is \(8\).
Convert \(\frac{1}{4}\) to an equivalent fraction with a denominator of \(8\):

• Multiply both the numerator and the denominator by \(2\).
• \(\frac{1}{4}\) becomes \(\frac{2}{8}\).

Now, it's easy to add \(\frac{2}{8}\) and \(\frac{1}{8}\):

• Simply add the numerators: \(2 + 1 = 3\).
• The sum is \(\frac{3}{8}\).

Choosing the right approach simplifies the addition of fractions and is a skill often needed when working with expressions involving several terms.

Mathematical Terms

Understanding mathematical terms can significantly boost your problem-solving capability. In this context, a term refers to a single mathematical expression that can be a constant, a variable, or a product of both.
Consider an expression like \(2^{-2} + 2^{-3}\). Here, both \(2^{-2}\) and \(2^{-3}\) are individual terms. Each term must be evaluated or simplified separately before performing operations such as addition or subtraction.
Terms can be progressively complex, sometimes involving exponents, variables, and other mathematical functions. Knowing how to correctly interpret and manipulate each term is crucial for accurately solving and simplifying expressions.
By breaking down the problem and understanding each term's role, you gain a clearer perspective, making it easier to apply mathematical rules and reach the solution efficiently.