Problem 108 - College Algebra

Question

The algebraic expression \(152 a^{-1 / 5}\) describes the percentage of U.S. taxpayers who are \(a\) years old who file early. Evaluate the algebraic expression for \(a=32\) Describe what the answer means in practical terms.

Step-by-Step Solution

Answer

The result derived from the algebraic expression will equate to the percentage of 32-year-old taxpayers who are expected to file early in the United States, and this result must be correctly interpreted in this practical context.

Step 1: Substitution

Substitute the value of \(a\) in the given algebraic expression. So, for \(a = 32\), the expression becomes \(152 \times 32^{-1/5}\).

Step 2: Simplification

Simplify the expression where \(32^{-1/5}\) represents the fifth root of 32 inverted. Calculate that value first and then multiply by 152 to get a numerical value, represented as a percentage.

Step 3: Interpretation

After calculating, translate the decimal to a percentage and interpret the result in the context of the problem, specifically what this percentage means about 32-year-old U.S. taxpayers filing early.

Key Concepts

Percentage Calculations

Calculating percentages involves understanding how a number relates to 100. It's like finding out what fraction of a whole number our result represents.
For instance, in the problem we are solving, after finding the numerical value from the expression and simplification steps, we get a decimal that shows the proportion of a whole number.
This decimal needs to be converted into a percentage by multiplying by 100.

• First, perform any operations involving fractions in the expression to arrive at a simplified number.
• Then, convert the simplified decimal result into a percentage.

For example, if our calculation gives us 0.25 after simplification, it would mean 25% after converting it to a percentage by multiplying by 100. This makes it easier to visualize and understand in terms of parts per hundred.

Exponent Rules

Exponents are a way of expressing repeated multiplication of a number by itself. In the exercise, the expression features an exponent of \(-1/5\).
This exponent indicates two operations:

• The negative sign means you will take the reciprocal of the base.
• The fraction \(1/5\) indicates you will find the fifth root of the number.

For example, with \(32^{-1/5}\):

• First, consider \(32^{1/5}\), which is the fifth root of 32.
• Next, take the reciprocal because of the negative sign, resulting in \(1/(\text{fifith root of 32})\).

Understanding these rules makes it easier to simplify expressions involving exponents and helps manipulate complex algebraic expressions accurately.

Practical Interpretation

Once the mathematical portion of the exercise has been solved and the simplified expression has been converted to a percentage, interpreting the result provides valuable insights.
In our exercise, this figure represents the percentage of 32-year-old U.S. taxpayers who file their taxes early.
Seeing the percentage allows us to easily comprehend the real-world aspect of the calculation.

• For example, if after calculating and converting, the percentage is 40%, it means that 40 out of every 100 taxpayers aged 32 file early.
• This interpretation can help in understanding trends and behaviors within specific age groups related to tax filing.

By translating numerical results into realistic scenarios, we appreciate how algebraic expressions can depict everyday situations.