Problem 38 - Algebra for College Students

Question

Simplify each numerical expression. \(2^{-4}+5^{-1}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{21}{80}\).

Step 1: Recognize and Use the Negative Exponent Rule

When dealing with negative exponents, remember that a negative exponent means the reciprocal of the base raised to the opposite positive exponent. Thus, \(a^{-n} = \frac{1}{a^n}\). In our expression, apply this rule: \(2^{-4} = \frac{1}{2^4}\) and \(5^{-1} = \frac{1}{5^1}\).

Step 2: Calculate the Positive Exponents

Now, calculate the values of the positive exponents. For \(2^4\), multiplying 2 by itself four times gives us 16. Hence, \(\frac{1}{2^4} = \frac{1}{16}\). For \(5^1\), we know 5 raised to the power of 1 is just 5. So, \(\frac{1}{5^1} = \frac{1}{5}\).

Step 3: Add the Fractions

Now, add the two fractions from Step 2: \(\frac{1}{16} + \frac{1}{5}\). To do so, find a common denominator, which is 80 in this case. Convert each fraction: \(\frac{1}{16} = \frac{5}{80}\) and \(\frac{1}{5} = \frac{16}{80}\). Add them together: \(\frac{5}{80} + \frac{16}{80} = \frac{21}{80}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a messy desk; it makes things clearer and easier to understand. In mathematics, simplifying involves reducing expressions to their simplest form. By doing this, you make calculations easier and results clearer.

• Identify the parts of the expression that can be simplified.
• Apply mathematical rules and operations correctly.
• Combine like terms to make expressions as short and straightforward as possible.

When you simplify, always keep in mind the mathematical operations and rules. With each step, check to ensure the expression is moving closer to being its simplest form. This approach helps not just with calculations, but also with a deeper understanding of the relationships between numbers and operations.

Fractions

Fractions are a way to represent numbers that are not whole. They consist of a numerator (top part) and a denominator (bottom part). Understanding fractions is crucial when dealing with problems involving parts of a whole, such as adding or subtracting portions of pizza or, in our context, combining fractions.

• A fraction represents a division operation. For example, \( \frac{1}{4} \) means 1 divided by 4.
• Common denominators are needed for adding and subtracting fractions. This means the bottoms of both fractions must be the same.

In our exercise, adding \( \frac{1}{16} + \frac{1}{5} \) required us to find a common denominator. Without this step, it’s like adding apples to oranges. By converting both fractions to have a denominator of 80, we could easily add them together. Remember to always simplify your final fraction if possible to keep it neat and tidy.

Exponent Rules

Exponent rules let us work with powers more easily by providing guidelines on how to handle them. Negative exponents are an important part of this, often misunderstood at first glance. However, they're simply a way of expressing inverse relationships.

• A negative exponent means 'take the reciprocal.' For example, \( a^{-n} = \frac{1}{a^n} \).
• Always convert negative exponents to positive exponents by using multiplication with their reciprocal.

In our problem, the expressions \( 2^{-4} \) and \( 5^{-1} \) use this rule by converting them to \( \frac{1}{2^4} \) and \( \frac{1}{5^1} \) respectively. This step is crucial in breaking down complex expressions into manageable calculations. Mastering these rules not just helps in solving problems, but also builds a solid foundation for higher mathematical concepts.