91Ó°ÊÓ

Radical expressions involve roots, like square roots or cube roots. In the exercise given, we are dealing with a square root. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 × 3 = 9.
When we see an expression such as \( x^{-1/2} \), it can be rewritten as \( \frac{1}{\sqrt{x}}\). This transformation makes it easier to work with because it changes an exponent-based expression into a division problem. Understanding how to manipulate and simplify radical expressions is a key skill in algebra. Always remember to simplify the roots if they can be calculated exactly. For instance, \( \sqrt{4} = 2 \). This simplification helps in solving the problem step by step.

Substitution is a method used to simplify expressions and solve equations. It involves replacing a variable with a given number or another expression. In the exercise, we substitute \( x = 4 \) into the function \( f(x) = x^{-1/2} \). This step-by-step substitution looks like this:

• Start with the function: \( f(x) = x^{-1/2} \).
• Substitute 4 for \( x \): \( f(4) = 4^{-1/2} \).
• Rewrite using the radical expression: \( 4^{-1/2} = \frac{1}{\sqrt{4}} \).
• Simplify the square root: \( \sqrt{4} = 2 \).
• Final simplified form: \( f(4) = \frac{1}{2} \).

By substituting and simplifying, we make the problem easier to solve. This approach is particularly helpful in algebra and calculus, making complex problems more manageable.

Simplifying expressions involves reducing them to their most basic form. This makes them easier to understand and work with. In the problem provided, simplification occurs after substitution and involves calculating the root and reducing fractions. Here's how it works in this problem:

• After substitution, we have \( f(4) = \frac{1}{\sqrt{4}} \).
• Calculate \( \sqrt{4} \), which is 2.
• So, \( f(4) = \frac{1}{2} \).

In steps, we transformed \( 4^{-1/2} \) to \( \frac{1}{2} \) by calculating the square root and simplifying the fraction. Simplification also helps in solving equations faster and reducing errors. Ensuring accuracy in each step is crucial for achieving the right solution.