Sure! Let's break down the given expression step-by-step to understand how it can be solved:
The original expression to simplify is:
[tex]\[
\frac{5 \sqrt[3]{54}}{3 \sqrt[3]{40}}
\][/tex]
### Step 1: Calculate the cube roots
First, we need to calculate the cube root of 54 and the cube root of 40.
- Cube root of 54: [tex]\( \sqrt[3]{54} \approx 3.7797631496846193 \)[/tex]
- Cube root of 40: [tex]\( \sqrt[3]{40} \approx 3.4199518933533937 \)[/tex]
### Step 2: Multiply by respective constants
Next, we multiply the cube root of 54 by 5 and the cube root of 40 by 3 to get the numerator and the denominator respectively.
- Numerator: [tex]\( 5 \times 3.7797631496846193 \approx 18.898815748423097 \)[/tex]
- Denominator: [tex]\( 3 \times 3.4199518933533937 \approx 10.25985568006018 \)[/tex]
### Step 3: Form the fraction
Now, we form the fraction with the calculated numerator and denominator:
[tex]\[
\frac{18.898815748423097}{10.25985568006018}
\][/tex]
### Step 4: Simplify the fraction
Finally, we perform the division to simplify the fraction:
[tex]\[
\frac{18.898815748423097}{10.25985568006018} \approx 1.8420157493201934
\][/tex]
### Conclusion
Therefore, the simplified result of the given expression [tex]\( \frac{5 \sqrt[3]{54}}{3 \sqrt[3]{40}} \)[/tex] is approximately:
[tex]\[
1.8420157493201934
\][/tex]
