Simplified Base of the Function

Oct 13, 2025 by orovajewels.com 52 views

What is the Simplified Base of the Function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ ?

In mathematics, functions are a crucial concept that helps us describe and analyze various mathematical relationships. One of the essential aspects of functions is their representation in a simplified form. In this article, we will focus on simplifying the base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ . We will explore the properties of cube roots and simplify the given expression to its most basic form.

Before we dive into simplifying the function, let's take a closer look at cube roots. A cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. In mathematical notation, this is represented as $\sqrt[3]{x}$ . For example, $\sqrt[3]{8}=2$ because $2^3=8$ .

Now that we have a basic understanding of cube roots, let's simplify the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ . To simplify this expression, we need to find the cube root of 108 and then multiply it by $\frac{1}{4}$ .

Step 1: Find the Cube Root of 108

The cube root of 108 can be found by factoring 108 into its prime factors. We can write 108 as $2^2 \times 3^3$ . Since we are looking for the cube root, we can take the cube root of each prime factor separately.

$\sqrt[3]{108}=\sqrt[3]{2^2 \times 3^3}=\sqrt[3]{2^2} \times \sqrt[3]{3^3}=2 \times 3=6$

Step 2: Multiply by $\frac{1}{4}$

Now that we have found the cube root of 108, we can multiply it by $\frac{1}{4}$ to get the simplified form of the function.

$f(x)=\frac{1}{4}(\sqrt[3]{108})=\frac{1}{4} \times 6=\frac{6}{4}=\frac{3}{2}$

However, we are asked to find the simplified base of the function, not the simplified form of the function. To find the simplified base, we need to express the cube root of 108 in its simplest form.

Simplified Base of the Function

We can rewrite the cube root of 108 as $\sqrt[3]{108}=\sqrt[3]{2^2 \times 3^3}=2 \times 3 \times \sqrt[3]{3}=6 \times \sqrt[3]{3}$ .

Therefore, the simplified base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ is $6 \times \sqrt[3]{3}$ .

In this article, we simplified the base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ . We found the cube root of 108 and then multiplied it by $\frac{1}{4}$ to get the simplified form of the function. However, we were asked to find the simplified base of the function, not the simplified form. We expressed the cube root of 108 in its simplest form and found that the simplified base of the function is $6 \times \sqrt[3]{3}$ .

The simplified base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ is $6 \times \sqrt[3]{3}$ .

The final answer is: $\boxed{6 \sqrt[3]{3}}$
Q&A: Simplified Base of the Function $f(x)=\frac{1}{4}(\sqrt[3]{108})$

In our previous article, we simplified the base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ . We found that the simplified base of the function is $6 \times \sqrt[3]{3}$ . In this article, we will answer some frequently asked questions related to the simplified base of the function.

Q: What is the simplified base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ ?

A: The simplified base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ is $6 \times \sqrt[3]{3}$ .

Q: How do you find the simplified base of a function?

A: To find the simplified base of a function, you need to express the cube root of the number in its simplest form. This involves factoring the number into its prime factors and then taking the cube root of each prime factor separately.

Q: What is the cube root of 108?

A: The cube root of 108 can be found by factoring 108 into its prime factors. We can write 108 as $2^2 \times 3^3$ . Since we are looking for the cube root, we can take the cube root of each prime factor separately.

$\sqrt[3]{108}=\sqrt[3]{2^2 \times 3^3}=\sqrt[3]{2^2} \times \sqrt[3]{3^3}=2 \times 3=6$

Q: How do you multiply a cube root by a fraction?

A: To multiply a cube root by a fraction, you need to multiply the cube root by the numerator of the fraction and then divide by the denominator of the fraction.

For example, $\frac{1}{4} \times \sqrt[3]{108}=\frac{\sqrt[3]{108}}{4}$ .

Q: What is the difference between the simplified form of a function and the simplified base of a function?

A: The simplified form of a function is the expression that results from simplifying the function using algebraic manipulations. The simplified base of a function is the expression that results from expressing the cube root of the number in its simplest form.

For example, the simplified form of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ is $\frac{6}{4}=\frac{3}{2}$ . However, the simplified base of the function is $6 \times \sqrt[3]{3}$ .

Q: Can you provide more examples of simplified bases of functions?

A: Yes, here are a few more examples of simplified bases of functions:

The simplified base of the function $f(x)=\frac{1}{2}(\sqrt[3]{216})$ is $6 \times \sqrt[3]{2}$ .
The simplified base of the function $f(x)=\frac{3}{4}(\sqrt[3]{729})$ is $9 \times \sqrt[3]{3}$ .

In this article, we answered some frequently asked questions related to the simplified base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})$ . We provided examples of how to find the simplified base of a function and how to multiply a cube root by a fraction. We also highlighted the difference between the simplified form of a function and the simplified base of a function.

The final answer is: $\boxed{6 \sqrt[3]{3}}$