Certainly! To find the product:
[tex]\[
\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9} \cdot x^2 \sqrt[3]{28 x^2} \cdot x^5 \sqrt[3]{28 x} \cdot 4 x^2 \sqrt[3]{3 x^2} \cdot 4 x^5 \sqrt[3]{3 x},
\][/tex]
we can evaluate each term under the product rule [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
### Step 1: Simplify the Cube Root Expressions
1. [tex]\(\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9} = \sqrt[3]{16 x^7 \cdot 12 x^9} = \sqrt[3]{192 x^{16}}\)[/tex].
2. [tex]\(x^2 \sqrt[3]{28 x^2}\)[/tex].
3. [tex]\(x^5 \sqrt[3]{28 x}\)[/tex].
4. [tex]\(4 x^2 \sqrt[3]{3 x^2}\)[/tex].
5. [tex]\(4 x^5 \sqrt[3]{3 x}\)[/tex].
### Step 2: Combine Each Term
Combine all simplified expressions under one cube root and other multiplications outside:
[tex]\[
\sqrt[3]{192 x^{16}} \cdot x^2 \sqrt[3]{28 x^2} \cdot x^5 \sqrt[3]{28 x} \cdot 4 x^2 \sqrt[3]{3 x^2} \cdot 4 x^5 \sqrt[3]{3 x}.
\][/tex]
### Step 3: Further Simplification Inside the Cube Root
Rewrite each term:
[tex]\[
\sqrt[3]{192 x^{16}} = 192^{1/3} x^{16/3},
\][/tex]
[tex]\[
x^2 \sqrt[3]{28 x^2} = x^2 \cdot 28^{1/3} x^{2/3} = 28^{1/3} x^{2 + 2/3} = 28^{1/3} x^{8/3},
\][/tex]
[tex]\[
x^5 \sqrt[3]{28 x} = x^5 \cdot 28^{1/3} x^{1/3} = 28^{1/3} x^{5 + 1/3} = 28^{1/3} x^{16/3},
\][/tex]
[tex]\[
4 x^2 \sqrt[3]{3 x^2} = 4 x^2 \cdot 3^{1/3} x^{2/3} = 4 \cdot 3^{1/3} x^{8/3},
\][/tex]
[tex]\[
4 x^5 \sqrt[3]{3 x} = 4 x^5 \cdot 3^{1/3} x^{1/3} = 4 \cdot 3^{1/3} x^{16/3}.
\][/tex]
### Step 4: Combine Like Terms
Group together similar terms, i.e., all constants and all [tex]\(x\)[/tex] terms:
[tex]\[
= \sqrt[3]{192} \left(\sqrt[3]{28} \right)^2 \left(x^{8/3}\right) \left(x^{8/3}\right) \cdot 4^2 x^{16/3},
\][/tex]
/[tex]\[
= 192^{1/3} \cdot 28^{2/3} \cdot x^{16/3} \cdot 16 \cdot x^{16/3},
\][/tex]
### Step 5: Simplify the Constants and the Exponents
Combine exponents and constants:
[tex]\[
192^{1/3} \cdot 28^{2/3} \cdot 16 \cdot x^{16/3 + 16/3} = 192^{1/3} \cdot 28^{2/3} \cdot 16 \cdot x^{32/3}.
\][/tex]
Find and combine the value of constants ([tex]\(192\)[/tex] and [tex]\(28^2 \)[/tex]):
[tex]\[
192^{1/3} \approx 5.82, \quad 28^{2/3} \approx 7.287,
\][/tex]
Hence:
[tex]\[
5.82 \cdot 7.287 \cdot 16 \approx 676.20.
\][/tex]
Combine it with [tex]\(x\)[/tex]:
[tex]\[
676.20x^{32/3}.
\][/tex]
### Final Result
The complete product simplifies to:
[tex]\[
676.20 x^{10.\bar{6}},
\][/tex]
where the product and the coefficients are combined accordingly.
