Let's break down the expression step-by-step:
The given expression is [tex]\(2 \sqrt[3]{6} \cdot 5 \sqrt[3]{9}\)[/tex].
### Step 1: Understand the Components
- [tex]\(\sqrt[3]{6}\)[/tex] is the cubic root of 6.
- [tex]\(\sqrt[3]{9}\)[/tex] is the cubic root of 9.
- We are asked to multiply these cubic roots by the constants 2 and 5, respectively.
### Step 2: Calculate the Cubic Roots
- First, we find the value of the cubic root of 6:
[tex]\[
\sqrt[3]{6} \approx 1.817
\][/tex]
- Next, we find the value of the cubic root of 9:
[tex]\[
\sqrt[3]{9} \approx 2.080
\][/tex]
### Step 3: Multiply the Constants and the Cubic Roots
First, we simplify the multiplication inside the parentheses:
[tex]\[
2 \cdot \sqrt[3]{6} \approx 2 \cdot 1.817 \approx 3.634
\][/tex]
Next, we do the same for the other term:
[tex]\[
5 \cdot \sqrt[3]{9} \approx 5 \cdot 2.080 \approx 10.400
\][/tex]
Then, we multiply these two intermediate results together:
[tex]\[
3.634 \cdot 10.400 \approx 37.798
\][/tex]
### Step 4: Combine the Results
Hence, the value of the original expression [tex]\(2 \sqrt[3]{6} \cdot 5 \sqrt[3]{9}\)[/tex] is:
[tex]\[
37.798
\][/tex]
Moreover, another way to approach it is to recognize mathematical simplification:
[tex]\[
2 \cdot 5 \cdot \sqrt[3]{6 \cdot 9} = 10 \cdot \sqrt[3]{54}
\][/tex]
Where [tex]\(\sqrt[3]{54} \approx 3.780\)[/tex].
So multiplying 10 by [tex]\(\sqrt[3]{54}\)[/tex]:
[tex]\[
10 \cdot 3.780 \approx 37.798
\][/tex]
Thus, the final value is:
[tex]\[
2 \sqrt[3]{6} \cdot 5 \sqrt[3]{9} \approx 37.798
\][/tex]
