Certainly! Let's tackle this problem step-by-step.
Given the quotient to simplify:
[tex]\[ \frac{\sqrt[3]{60}}{\sqrt[3]{20}} \][/tex]
#### Step 1: Calculate the cube roots
We first need to find the cube roots of the numbers 60 and 20.
1. The cube root of 60:
[tex]\[ \sqrt[3]{60} \approx 3.9148676411688634 \][/tex]
2. The cube root of 20:
[tex]\[ \sqrt[3]{20} \approx 2.7144176165949063 \][/tex]
#### Step 2: Form the quotient
We then form the quotient by dividing the cube root of 60 by the cube root of 20:
[tex]\[ \frac{\sqrt[3]{60}}{\sqrt[3]{20}} = \frac{3.9148676411688634}{2.7144176165949063} \approx 1.4422495703074085 \][/tex]
#### Step 3: Simplify the quotient
Finally, we express the simplified quotient in a more recognizable form. This quotient can also be rewritten as:
[tex]\[ 2(\sqrt[3]{5}) = 2 \times \sqrt[3]{20} \][/tex]
where:
[tex]\[ 2 \times 2.7144176165949063 \approx 5.428835233189813 \][/tex]
Thus, the values we have are:
[tex]\[ \sqrt[3]{60} \approx 3.9148676411688634 \][/tex]
[tex]\[ \sqrt[3]{20} \approx 2.7144176165949063 \][/tex]
[tex]\[ \frac{\sqrt[3]{60}}{\sqrt[3]{20}} \approx 1.4422495703074085 \][/tex]
[tex]\[ 2 \times \sqrt[3]{20} \approx 5.428835233189813 \][/tex]
So, the given quotient
[tex]\[ \frac{\sqrt[3]{60}}{\sqrt[3]{20}} \approx 1.4422495703074085 \][/tex]
This is the simplified form of the quotient given in the problem.
Hopefully, this clarifies the solution for you! If you need further assistance, feel free to ask.
