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Simplify [tex](\sqrt{2})(\sqrt[3]{2})[/tex].

A. [tex]2^{\frac{1}{6}}[/tex]
B. [tex]2^{\frac{2}{3}}[/tex]
C. [tex]2^{\frac{5}{6}}[/tex]
D. [tex]2^{\frac{7}{6}}[/tex]


Sagot :

To simplify the expression [tex]\((\sqrt{2})(\sqrt[3]{2})\)[/tex], we need to express both terms with the same base and deal with the exponents.

First, recall the meanings of the terms:
- [tex]\(\sqrt{2}\)[/tex] is the same as [tex]\(2^{1/2}\)[/tex].
- [tex]\(\sqrt[3]{2}\)[/tex] is the same as [tex]\(2^{1/3}\)[/tex].

The given product is:

[tex]\[ (\sqrt{2})(\sqrt[3]{2}) = (2^{1/2})(2^{1/3}) \][/tex]

When multiplying exponential expressions with the same base, we add their exponents:

[tex]\[ 2^{1/2} \times 2^{1/3} = 2^{1/2 + 1/3} \][/tex]

Next, we need to add the fractions [tex]\(1/2\)[/tex] and [tex]\(1/3\)[/tex]. To do this, we find a common denominator. The least common multiple of 2 and 3 is 6. Converting each fraction to have this common denominator:

[tex]\[ 1/2 = 3/6 \quad \text{and} \quad 1/3 = 2/6 \][/tex]

Now, adding these fractions together:

[tex]\[ \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \][/tex]

So the exponent sum is [tex]\(5/6\)[/tex], and the simplified product is:

[tex]\[ 2^{1/2 + 1/3} = 2^{5/6} \][/tex]

Therefore, the simplified form of [tex]\((\sqrt{2})(\sqrt[3]{2})\)[/tex] is:

[tex]\[ \boxed{2^{\frac{5}{6}}} \][/tex]