Problem 52 - College Algebra with Modeling and Visualization

Question

Evaluate \(f(x)\) at the given \(x\) Approximate each result to the nearest hundredth. $$ f(x)=x^{5 / 4}-x^{-3 / 4}, \quad x=7 $$

Step-by-Step Solution

Answer

The approximate value is 9.96.

Step 1: Substitute the Given Value

Substitute the given value \(x = 7\) into the function \(f(x) = x^{5/4} - x^{-3/4}\). This gives us: \[f(7) = 7^{5/4} - 7^{-3/4}\]

Step 2: Calculate the Positive Fractional Exponent

Calculate \(7^{5/4}\). We break it down into two steps: first, compute the fourth root of 7, then raise it to the power of 5. Use a calculator to find: \[\7^{5/4} \approx 10.3185\]

Step 3: Calculate the Negative Fractional Exponent

Calculate \(7^{-3/4}\). This is the same as \(\frac{1}{{7^{3/4}}}\). Find the fourth root of 7 and then raise it to the power of 3, and use a calculator to determine: \[\7^{-3/4} \approx 0.363\]

Step 4: Subtract the Exponents Results

Subtract the result from Step 3 from the result from Step 2 to find: \[\f(7) = 10.3185 - 0.363 \approx 9.9555\]

Step 5: Round to the Nearest Hundredth

Round the final result to the nearest hundredth, which gives us:\[\f(7) \approx 9.96\]

Key Concepts

Understanding Fractional Exponents

Fractional exponents can seem tricky, but they are just a way to express roots using exponents. When an exponent is a fraction, the denominator tells us the root to take, while the numerator indicates the power to which we raise the result. For instance, if we have an exponent of \(\frac{5}{4}\), this means "take the fourth root" and then "raise it to the power of five." This makes it much easier to visualize and calculate. In our problem, for the expression \(7^{5/4}\), we first calculate \(\sqrt[4]{7}\) and then raise this result to the power of five. This approach allows us to break down complex expressions into more manageable calculations.

The Role of Substitution in Function Evaluation

Substitution is a method used to simplify expressions and solve equations by replacing variables with their actual values. This makes it easier to carry out evaluations step-by-step. In the current problem, we substitute \(x = 7\) into the function \(f(x) = x^{5/4} - x^{-3/4}\). This gives us the specific expression \(f(7) = 7^{5/4} - 7^{-3/4}\), which we then evaluate. Substitution is a powerful technique because it transforms a general expression into something concrete that can be directly calculated. It essentially turns abstract math into arithmetic.

How to Properly Round Numbers

Rounding is the process of simplifying a number to a specific degree of precision. This is often necessary when dealing with long decimal numbers, which can be unwieldy. Our task in the problem is to round the result to the nearest hundredth. When rounding to the hundredths place, you look at the digit in the thousandths place:

• If it is 5 or greater, round the hundredths place up by one.
• If it is less than 5, keep the hundredths place the same.

For the result \(9.9555\), the thousandths digit is 5, which tells us to round up, resulting in \(9.96\). Rounding makes numbers easier to understand and use, especially when precision beyond a certain point isn't necessary.