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Chapter 5: Problem 29

Express using positive exponents and simplify, if possible. \(2^{-2}+4^{-1}\)

Short Answer

Expert verified
The simplified expression is \(\frac{1}{2}\).

Step by step solution

01

Express Negative Exponents as Positive

To convert each term with a negative exponent into a term with a positive exponent, recall that a negative exponent implies the reciprocal of the base raised to the absolute value of the exponent. Thus, convert \(2^{-2}\) to \(\frac{1}{2^2}\) and \(4^{-1}\) to \(\frac{1}{4}\).
02

Simplify Exponents

Now, calculate the powers: \(2^2 = 4\), so \(\frac{1}{2^2} = \frac{1}{4}\). There is no need to simplify \(\frac{1}{4}\) further since it's already in its simplest form.
03

Add the Fractions

We now have the addition \(\frac{1}{4} + \frac{1}{4}\). Since the denominators are the same, we can simply add the numerators. Compute \(\frac{1+1}{4} = \frac{2}{4}\).
04

Simplify the Resulting Fraction

The fraction \(\frac{2}{4}\) can be simplified by finding the greatest common divisor (GCD) of 2 and 4, which is 2. Divide the numerator and the denominator by their GCD: \(\frac{2}{4} = \frac{1}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Exponents
Working with positive exponents is much easier than dealing with negative ones. When you see an exponent, it tells you how many times to multiply the base by itself. For example, in the expression \(2^2\), this means you should multiply 2 by itself, resulting in \(2 \times 2 = 4\).
Exponents can sometimes be negative. When that happens, it indicates that you take the reciprocal, or the "flip," of the number and then apply the positive exponent. So, \(2^{-2}\) becomes \(\frac{1}{2^2}\).
This means you first flip the base to get \(\frac{1}{2}\), then square it, which gives you \(\frac{1}{4}\).
  • A positive exponent shows how many times to multiply.
  • Flip the base and use the positive value when the exponent is negative.
Simplifying Fractions
Simplifying fractions helps make expressions easier to handle. A fraction is simplified when the numerator and the denominator have no common factors other than 1. For instance, consider the fraction \(\frac{2}{4}\).
Both 2 and 4 can be divided by 2, which is their greatest common divisor (GCD). When you divide both the numerator and the denominator by this number, you get \(\frac{1}{2}\).
To check if a fraction can be simplified:
  • Find the GCD of the numerator and denominator.
  • Divide both by the GCD to simplify.
Simplifying makes calculations cleaner and is helpful, especially in more complex math problems.
Adding Fractions
Adding fractions might seem tricky, but it becomes simple when you break it down. The most crucial aspect is ensuring that the fractions have the same denominator. If they do, you can add the numerators directly, keeping the denominator as it is. In our exercise, both fractions are \(\frac{1}{4}\).
Since the denominators are the same, simply add the numerators: \(1 + 1\). This yields \(\frac{2}{4}\), which can then be simplified.
Here's a step-by-step approach to adding fractions:
  • Check if the denominators are the same.
  • Add the numerators if they are.
  • Simplify the resulting fraction if possible.
In cases where the denominators differ, you would need to find a common denominator before adding.

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Most popular questions from this chapter

a. Find two binomials whose product is a binomial. b. Find two binomials whose product is a trinomial. c. Find two binomials whose product is a four-term polynomial.

Perform the operations. $$ (4 b-8)^{2} $$

Perform the indicated operations. a. \(\left(2 b^{2} d\right)^{2}\) b. \(\left(2 b^{2}+d\right)^{2}\)

Perform each division. $$ \frac{24 x^{7}-32 x^{2}}{16 x^{3}} $$

Add: \(\frac{5}{12}+\frac{1}{4}\)
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