What Is The Simplified Base Of The Function $f(x)=\frac{1}{4}(\sqrt[3]{108})^x$?A. 3 B. $3 \sqrt[3]{4}$ C. $6 \sqrt[3]{3}$ D. 27

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

Understanding the Function

The given function is $f(x)=\frac{1}{4}(\sqrt[3]{108})^x$. To simplify the base of this function, we need to analyze the expression inside the parentheses, which is $\sqrt[3]{108}$. This expression represents the cube root of 108.

Breaking Down the Cube Root

To simplify the cube root of 108, we can break it down into its prime factors. The prime factorization of 108 is $2^2 \times 3^3$. Therefore, we can rewrite the cube root of 108 as $\sqrt[3]{2^2 \times 3^3}$.

Simplifying the Cube Root

Now, we can simplify the cube root of 108 by using the properties of exponents. When we take the cube root of a product, we can take the cube root of each factor separately. Therefore, we can rewrite the cube root of 108 as $\sqrt[3]{2^2} \times \sqrt[3]{3^3}$.

Evaluating the Cube Roots

The cube root of $2^2$ is $2^{2/3}$, and the cube root of $3^3$ is $3$. Therefore, we can simplify the cube root of 108 as $2^{2/3} \times 3$.

Simplifying the Expression

Now, we can simplify the expression $2^{2/3} \times 3$ by combining the two factors. We can rewrite this expression as $3 \times 2^{2/3}$.

Rewriting the Expression

To rewrite the expression $3 \times 2^{2/3}$ in a more simplified form, we can use the properties of exponents. When we multiply two numbers with the same base, we can add their exponents. Therefore, we can rewrite the expression as $3 \times 2^{2/3} = 3 \times 2^{2/3} \times 2^{-2/3}$.

Simplifying the Expression

Now, we can simplify the expression $3 \times 2^{2/3} \times 2^{-2/3}$ by combining the two factors with the same base. We can rewrite this expression as $3 \times 2^{2/3-2/3}$.

Evaluating the Exponents

The exponent $2/3-2/3$ is equal to 0. Therefore, we can simplify the expression as $3 \times 2^0$.

Simplifying the Expression

Now, we can simplify the expression $3 \times 2^0$ by evaluating the exponent. The exponent 0 is equal to 1. Therefore, we can rewrite the expression as $3 \times 1$.

Evaluating the Expression

The expression $3 \times 1$ is equal to 3. Therefore, we can simplify the base of the function $f(x)=\frac{1}{4}(\sqrt[3]{108})^x$ as 3.

Conclusion

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

References

• [1] Khan Academy. (n.d.). Exponents and Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponents/x2f-exponents-and-exponential-functions/x2f-exponents-and-exponential-functions/v/exponents-and-exponential-functions
• [2] Math Open Reference. (n.d.). Cube Root. Retrieved from https://www.mathopenref.com/cuberoot.html

Discussion

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

A. 3 B. $3 \sqrt[3]{4}$ C. $6 \sqrt[3]{3}$ D. 27

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

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

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

Q: Why is the simplified base of the function 3?

A: The simplified base of the function is 3 because the cube root of 108 can be simplified as $3 \times 2^{2/3}$. When we multiply two numbers with the same base, we can add their exponents. Therefore, we can rewrite the expression as $3 \times 2^{2/3} = 3 \times 2^{2/3} \times 2^{-2/3}$. The exponent $2/3-2/3$ is equal to 0, so we can simplify the expression as $3 \times 2^0$. The exponent 0 is equal to 1, so we can rewrite the expression as $3 \times 1$. The expression $3 \times 1$ is equal to 3.

Q: How do we simplify the cube root of 108?

A: To simplify the cube root of 108, we can break it down into its prime factors. The prime factorization of 108 is $2^2 \times 3^3$. Therefore, we can rewrite the cube root of 108 as $\sqrt[3]{2^2 \times 3^3}$. We can simplify the cube root of 108 by using the properties of exponents. When we take the cube root of a product, we can take the cube root of each factor separately. Therefore, we can rewrite the cube root of 108 as $\sqrt[3]{2^2} \times \sqrt[3]{3^3}$.

Q: What is the cube root of $2^2$ and $3^3$?

A: The cube root of $2^2$ is $2^{2/3}$, and the cube root of $3^3$ is $3$.

Q: How do we simplify the expression $3 \times 2^{2/3}$?

A: We can simplify the expression $3 \times 2^{2/3}$ by combining the two factors with the same base. We can rewrite the expression as $3 \times 2^{2/3} = 3 \times 2^{2/3} \times 2^{-2/3}$. The exponent $2/3-2/3$ is equal to 0, so we can simplify the expression as $3 \times 2^0$. The exponent 0 is equal to 1, so we can rewrite the expression as $3 \times 1$. The expression $3 \times 1$ is equal to 3.

Q: What is the final simplified base of the function?

A: The final simplified base of the function is 3.

Conclusion

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

References

• [1] Khan Academy. (n.d.). Exponents and Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponents/x2f-exponents-and-exponential-functions/x2f-exponents-and-exponential-functions/v/exponents-and-exponential-functions
• [2] Math Open Reference. (n.d.). Cube Root. Retrieved from https://www.mathopenref.com/cuberoot.html

Discussion

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

A. 3 B. $3 \sqrt[3]{4}$ C. $6 \sqrt[3]{3}$ D. 27

The correct answer is A. 3.