To solve for the product [tex]\(\sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32}\)[/tex], let's break it down step by step.
1. Convert the radicals to exponents:
- [tex]\(\sqrt[3]{2}\)[/tex] is equivalent to [tex]\(2^{\frac{1}{3}}\)[/tex].
- [tex]\(\sqrt[4]{2}\)[/tex] is equivalent to [tex]\(2^{\frac{1}{4}}\)[/tex].
- [tex]\(\sqrt[12]{32}\)[/tex] is equivalent to [tex]\(32^{\frac{1}{12}}\)[/tex].
2. Simplify [tex]\(32^{\frac{1}{12}}\)[/tex] using the fact that [tex]\(32\)[/tex] can be written as [tex]\(2^5\)[/tex]:
- [tex]\(32 = 2^5\)[/tex].
- Therefore, [tex]\(32^{\frac{1}{12}} = (2^5)^{\frac{1}{12}} = 2^{\frac{5}{12}}\)[/tex].
3. Combine the exponents since the bases are the same:
- [tex]\(2^{\frac{1}{3}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{5}{12}}\)[/tex].
4. Add the exponents:
[tex]\[
\frac{1}{3} + \frac{1}{4} + \frac{5}{12}
\][/tex]
To add these fractions, find a common denominator. The common denominator for 3, 4, and 12 is 12.
- [tex]\(\frac{1}{3} = \frac{4}{12}\)[/tex]
- [tex]\(\frac{1}{4} = \frac{3}{12}\)[/tex]
- [tex]\(\frac{5}{12} = \frac{5}{12}\)[/tex]
Now, add these fractions together:
[tex]\[
\frac{4}{12} + \frac{3}{12} + \frac{5}{12} = \frac{4 + 3 + 5}{12} = \frac{12}{12} = 1
\][/tex]
5. Combine into a single base:
- This means the combined exponent is 1, so [tex]\(2^1 = 2\)[/tex].
Therefore, the product [tex]\(\sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32}\)[/tex] equals 2. The correct answer is:
b. 2
