To solve the given problem, we need to find the simplified product:
[tex]\[ \sqrt[3]{2 x^5} \cdot \sqrt[3]{64 x^9} \][/tex]
Let's break this down step by step:
1. Express the cube roots separately:
[tex]\[ \sqrt[3]{2 x^5} \][/tex]
[tex]\[ \sqrt[3]{64 x^9} \][/tex]
2. Combine the expressions under a single cube root:
[tex]\[ \sqrt[3]{(2 x^5) \cdot (64 x^9)} \][/tex]
3. Multiply the terms inside the cube root:
[tex]\[ = \sqrt[3]{2 \cdot 64 \cdot x^5 \cdot x^9} \][/tex]
4. Simplify the constants and the exponents of [tex]\(x\)[/tex]:
We know [tex]\(64 = 2^6\)[/tex], so:
[tex]\[ 2 \cdot 64 = 2 \cdot 2^6 = 2^7 \][/tex]
For the [tex]\(x\)[/tex] terms:
[tex]\[ x^5 \cdot x^9 = x^{5+9} = x^{14} \][/tex]
5. Combine the simplified constants and exponents:
[tex]\[ = \sqrt[3]{2^7 x^{14}} \][/tex]
6. Apply the cube root to each part:
The cube root of [tex]\(2^7\)[/tex] is:
[tex]\[ \sqrt[3]{2^7} = 2^{7/3} \][/tex]
And the cube root of [tex]\(x^{14}\)[/tex] is:
[tex]\[ \sqrt[3]{x^{14}} = x^{14/3} \][/tex]
7. Recombine the terms:
[tex]\[ \sqrt[3]{2^7 x^{14}} = 2^{7/3} x^{14/3} \][/tex]
8. Identify the simplified form among the given choices:
From the given choices, identify the expression that matches [tex]\(2^{7/3} x^{14/3}\)[/tex]:
- [tex]\( 8 x^{63} \sqrt{2 x^2} \)[/tex]
- [tex]\( 2 x^5 \sqrt[3]{8 x^4} \)[/tex]
- [tex]\( 4 x \sqrt[43]{2 x^2} \)[/tex]
- [tex]\( 8 x_4^4 \sqrt{2 x^2} \)[/tex]
Comparing, the correct simplified form corresponds to the simplified solution, which is choice 3.
Therefore, the correct answer is:
[tex]\[ 3 \][/tex]
