Whether you're a student or a professional, IDNLearn.com has answers for everyone. Ask any question and receive timely, accurate responses from our dedicated community of experts.
Sagot :
To determine the simplified base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex], let's proceed step-by-step:
1. Calculate the cube root of 108:
[tex]\[ \sqrt[3]{108} \approx 4.762203155904598 \][/tex]
2. Express 108 as a product of prime factors:
[tex]\[ 108 = 2^2 \cdot 3^3 \][/tex]
3. Find the cube root of 108 in terms of its prime factors:
[tex]\[ \sqrt[3]{108} = \sqrt[3]{2^2 \cdot 3^3} \][/tex]
This can be broken down as:
[tex]\[ \sqrt[3]{2^2} = 2^{2/3} \][/tex]
and
[tex]\[ \sqrt[3]{3^3} = 3 \][/tex]
Therefore,
[tex]\[ \sqrt[3]{108} = 2^{2/3} \cdot 3 \][/tex]
4. Simplify the base of the function:
[tex]\[ \text{Base} = \frac{1}{4} \sqrt[3]{108} \][/tex]
Substitute [tex]\(\sqrt[3]{108}\)[/tex] with its simplified form:
[tex]\[ \text{Base} = \frac{1}{4} (2^{2/3} \cdot 3) \][/tex]
5. Combine and simplify the expression:
[tex]\[ \text{Base} = \frac{1}{4} \cdot 2^{2/3} \cdot 3 \][/tex]
Since [tex]\(\frac{1}{4} = 2^{-2}\)[/tex], we can write:
[tex]\[ \text{Base} = 2^{-2} \cdot 2^{2/3} \cdot 3 = 2^{2/3 - 2} \cdot 3 \][/tex]
Simplifying [tex]\(2^{2/3 - 2}\)[/tex]:
[tex]\[ 2^{2/3 - 2} = 2^{2/3 - 6/3} = 2^{-4/3} \][/tex]
So,
[tex]\[ \text{Base} = 2^{-4/3} \cdot 3 \][/tex]
6. Converting the base to a decimal form:
Using the numerical result, we have:
[tex]\[ 2^{-4/3} \cdot 3 \approx 0.520020955762976 \][/tex]
Therefore, the simplified base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex] is approximately [tex]\( 0.520020955762976 \)[/tex]. This results from a combination of the cube root of 108 and the fraction [tex]\(\frac{1}{4}\)[/tex].
1. Calculate the cube root of 108:
[tex]\[ \sqrt[3]{108} \approx 4.762203155904598 \][/tex]
2. Express 108 as a product of prime factors:
[tex]\[ 108 = 2^2 \cdot 3^3 \][/tex]
3. Find the cube root of 108 in terms of its prime factors:
[tex]\[ \sqrt[3]{108} = \sqrt[3]{2^2 \cdot 3^3} \][/tex]
This can be broken down as:
[tex]\[ \sqrt[3]{2^2} = 2^{2/3} \][/tex]
and
[tex]\[ \sqrt[3]{3^3} = 3 \][/tex]
Therefore,
[tex]\[ \sqrt[3]{108} = 2^{2/3} \cdot 3 \][/tex]
4. Simplify the base of the function:
[tex]\[ \text{Base} = \frac{1}{4} \sqrt[3]{108} \][/tex]
Substitute [tex]\(\sqrt[3]{108}\)[/tex] with its simplified form:
[tex]\[ \text{Base} = \frac{1}{4} (2^{2/3} \cdot 3) \][/tex]
5. Combine and simplify the expression:
[tex]\[ \text{Base} = \frac{1}{4} \cdot 2^{2/3} \cdot 3 \][/tex]
Since [tex]\(\frac{1}{4} = 2^{-2}\)[/tex], we can write:
[tex]\[ \text{Base} = 2^{-2} \cdot 2^{2/3} \cdot 3 = 2^{2/3 - 2} \cdot 3 \][/tex]
Simplifying [tex]\(2^{2/3 - 2}\)[/tex]:
[tex]\[ 2^{2/3 - 2} = 2^{2/3 - 6/3} = 2^{-4/3} \][/tex]
So,
[tex]\[ \text{Base} = 2^{-4/3} \cdot 3 \][/tex]
6. Converting the base to a decimal form:
Using the numerical result, we have:
[tex]\[ 2^{-4/3} \cdot 3 \approx 0.520020955762976 \][/tex]
Therefore, the simplified base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex] is approximately [tex]\( 0.520020955762976 \)[/tex]. This results from a combination of the cube root of 108 and the fraction [tex]\(\frac{1}{4}\)[/tex].
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.