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Magazine Chapter 0: Problem 21
Evaluate each expression. $$ 2^{-2}+2^{-3} $$
Short Answer
Expert verified
\(\frac{3}{8}\)
Step by step solution
01
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, \(2^{-n} = \frac{1}{2^n}\).
02
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 \(\frac{1}{2^3}\).
03
Calculate the powers of 2
Calculate \(2^2\) and \(2^3\). We have \(2^2 = 4\) and \(2^3 = 8\).
04
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 \(\frac{1}{8}\).
05
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 \(\frac{2}{8} + \frac{1}{8} = \frac{3}{8}\).
06
Final result
The simplified result of the expression \(2^{-2} + 2^{-3}\) is \(\frac{3}{8}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
For any positive integer n, \(2^n\) refers to multiplying 2 by itself n times.Key powers of 2 used frequently:
- \(2^0 = 1\)
- \(2^1 = 2\)
- \(2^2 = 4\)
- \(2^3 = 8\)
- \(2^4 = 16\)
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:
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: \(a^{-n} = \frac{1}{a^n}\)
