Problem 50 - A Graphical Approach to Precalculus with Limits

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 of \( f(7) \) is 10.72.

Step 1: Substitute x into the function

The given function is \( f(x) = x^{5/4} - x^{-3/4} \). We need to evaluate this function at \( x = 7 \). So, substitute 7 into the function to get \( f(7) = 7^{5/4} - 7^{-3/4} \).

Step 2: Evaluate \( 7^{5/4} \)

Compute \( 7^{5/4} \). This is equivalent to the fourth root of \( 7^5 \). First, calculate \( 7^5 = 16807 \) and then find the fourth root, which is approximately \( 10.96 \).

Step 3: Evaluate \( 7^{-3/4} \)

Compute \( 7^{-3/4} \), which is the same as \( \frac{1}{7^{3/4}} \). Calculate \( 7^3 = 343 \) and then the fourth root of 343 is approximately \( 4.18 \). Thus, \( 7^{-3/4} \) is approximately \( \frac{1}{4.18} = 0.24 \).

Step 4: Subtract the results

Subtract the result from Step 3 from the result in Step 2. Therefore, calculate \( 10.96 - 0.24 = 10.72 \).

Step 5: Round the result

The result \( 10.72 \) is already to the nearest hundredth, so no further rounding is necessary.

Key Concepts

Exponents

In mathematics, exponents are a way to express repeated multiplication of the same number. They appear frequently in algebra and calculus. When we write numbers in the form of powers, like in the function given, they are said to have exponents. For example, in the term \( x^{5/4} \), the exponent is \( \frac{5}{4} \). This means we are raising \( x \) to a power that involves both multiplication and a root.

Exponents can have fractional values, as we see in problems like this. A fraction in the exponent, such as \( 5/4 \), indicates that we multiply the base (here, \( x \)) by itself a number of times given by the numerator (5), and then take a root of that result, the degree of which is the denominator (4). This combination of power and root can initially seem complicated, but by understanding the role of each part, it becomes quite manageable.

• The numerator of the exponent determines how many times to multiply the base by itself.
• The denominator tells you the root you need to take of that result.

Substitution

Substitution is a method used to evaluate functions for given specific values. It involves replacing the variables in a function with numbers to compute the expression. In the problem from the original step by step solution, we are tasked with substituting into function \( f(x) \) at \( x = 7 \).

By substituting, we take the place of \( x \) in the equation and replace it with 7, turning \( f(x) = x^{5/4} - x^{-3/4} \) into \( f(7) = 7^{5/4} - 7^{-3/4} \). This allows us to then focus on solving this more straightforward arithmetic problem by applying arithmetic operations appropriate for exponents.

Substitution is a crucial step because it helps bridge the gap between abstract algebraic expressions and concrete numerical evaluations, making complex functions more accessible to solve.

Roots

Roots are an essential concept in mathematics, often working hand in hand with exponents. When you see an expression like \( x^{5/4} \), the exponent is indicating not just an exponentiation but also a root to be taken.

In the example given, calculating \( 7^{5/4} \) first involves calculating \( 7^5 \) and then taking the fourth root of that result. Similarly, for \( 7^{-3/4} \), you'll need to compute \( 7^3 \) and then find the fourth root, which is the inverse process, as indicated by the negative sign.

The idea of roots can be initially confusing, especially when dealing with fractional exponents.

• The root (e.g., fourth root) is always determined by the denominator of the exponent.
• Taking roots is like undoing exponentiation, akin to how division undoes multiplication.

Mathematical Approximation

Mathematical approximation becomes necessary when dealing with complex calculations that do not result in simple whole numbers. Rounding the results can create a more manageable number to work with, especially when precision is only needed to a certain place value.

In this exercise, rounding to the nearest hundredth is asked. After evaluating \( 7^{5/4} \) and \( 7^{-3/4} \), the outcomes were approximate to \( 10.96 \) and \( 0.24 \) respectively. By subtracting these, we get \( 10.72 \), already presented in a rounded form.

Approximation helps when results from calculations are irrational or unwieldy due to roots or non-integer powers. It helps make the numbers more understandable and useful, emphasizing the balance between exactitude and practicality in mathematical problem-solving.