To solve the expression [tex]\(\sqrt[3]{2} \times \sqrt[4]{2} \times \sqrt[12]{32}\)[/tex], let's break it down step by step.
1. Convert the root expressions to powers of 2:
- The cube root of 2, [tex]\(\sqrt[3]{2}\)[/tex], can be written as [tex]\(2^{1/3}\)[/tex].
- The fourth root of 2, [tex]\(\sqrt[4]{2}\)[/tex], can be written as [tex]\(2^{1/4}\)[/tex].
2. Simplify the term [tex]\(\sqrt[12]{32}\)[/tex]:
- We know that [tex]\(32\)[/tex] can be expressed as [tex]\(2^5\)[/tex] because [tex]\(32 = 2 \times 2 \times 2 \times 2 \times 2.\)[/tex]
- Therefore, [tex]\(\sqrt[12]{32}\)[/tex] can be rewritten as [tex]\(\sqrt[12]{2^5} = (2^5)^{1/12} = 2^{5/12}.\)[/tex]
3. Combine the expressions using properties of exponents:
- We have [tex]\(\sqrt[3]{2} \times \sqrt[4]{2} \times \sqrt[12]{32} = 2^{1/3} \times 2^{1/4} \times 2^{5/12}.\)[/tex]
- When multiplying expressions with the same base, we add the exponents: [tex]\(2^{1/3 + 1/4 + 5/12}.\)[/tex]
4. Find the common denominator and add the exponents:
- The common denominator of 3, 4, and 12 is 12.
- Convert each fraction to have a denominator of 12:
- [tex]\(1/3 = 4/12\)[/tex]
- [tex]\(1/4 = 3/12\)[/tex]
- [tex]\(5/12\)[/tex] already has a denominator of 12.
Adding these fractions together gives:
[tex]\[
\frac{1}{3} + \frac{1}{4} + \frac{5}{12} = \frac{4}{12} + \frac{3}{12} + \frac{5}{12} = \frac{4 + 3 + 5}{12} = \frac{12}{12} = 1.
\][/tex]
5. Combine the terms:
- This simplifies to [tex]\(2^1\)[/tex], which is simply 2.
Therefore, the final expression evaluates to:
[tex]\[
\sqrt[3]{2} \times \sqrt[4]{2} \times \sqrt[12]{32} = 2.
\][/tex]
