91Ó°ÊÓ

In calculus and algebra, evaluating a function means finding the value of the function at a specific input. This is a fundamental concept where a given value, generally represented as \( x \), is plugged into the function's formula, letting us see what the function outputs.

Consider the example of the function given: \( h(x) = x^{1/4} \). In the exercise, we need to evaluate this function at \( x = 81 \). To do this:

• Identify the function definition, \( h(x) = x^{1/4} \).
• Replace \( x \) with 81, giving us \( h(81) = 81^{1/4} \).

This process of function evaluation is crucial for understanding how functions behave for specific inputs.

Exponents and roots are key elements in understanding and simplifying complex mathematical expressions. Knowing how they work helps us interpret mathematical functions and carry out operations on them efficiently.

An exponent refers to the power a number is raised to. For example, in \( 3^4 \), the number 3 is the "base," and 4 is the "exponent," meaning 3 is multiplied by itself four times.
A root is essentially the inverse operation of an exponent. The root asks what number can be multiplied by itself a specific number of times to reach another number. The fourth root of 81 can be written as \( 81^{1/4} \).

• To calculate \( 81^{1/4} \), think about what number raised to the power of 4 would equal 81.
• Recognizing that \( 3^4 = 81 \), the fourth root of 81 is 3.

Roots are fundamental for solving equations and transforming expressions in calculus and algebra.

A step-by-step solution in mathematics is incredibly helpful for building understanding and seeing exactly how a problem is solved from start to finish. Let's revisit how this approach is applied in the given exercise:

1. **Substitute the Value:**

• We start by substituting the value into the function, i.e., plug 81 into \( h(x) = x^{1/4} \), which gives \( h(81) = 81^{1/4} \).

2. **Understand the Operation:**

• The expression \( 81^{1/4} \) requires finding the fourth root of 81. This involves determining what number can be multiplied by itself four times to make 81.

3. **Calculate the Result:**

• Identifying that \( 3^4 = 81 \), we find \( 81^{1/4} = 3 \).

Step-by-step solutions break down complex problems into manageable bites, making them approachable and easier to solve.