Let's solve the problem step-by-step to determine which of the given expressions is equivalent to the expression [tex]\(27^2 \times 81^{0.5}\)[/tex].
First, break down the expression [tex]\(27^2 \times 81^{0.5}\)[/tex].
1. Evaluate [tex]\(27^2\)[/tex]:
Note that [tex]\(27\)[/tex] can be expressed as [tex]\(3^3\)[/tex]:
[tex]\[
27 = 3^3
\][/tex]
Therefore,
[tex]\[
27^2 = (3^3)^2 = 3^{3 \times 2} = 3^6
\][/tex]
2. Evaluate [tex]\(81^{0.5}\)[/tex]:
Note that [tex]\(81\)[/tex] can be expressed as [tex]\(3^4\)[/tex]:
[tex]\[
81 = 3^4
\][/tex]
Therefore,
[tex]\[
81^{0.5} = (3^4)^{0.5} = 3^{4 \times 0.5} = 3^2
\][/tex]
3. Combine the results:
Now we have:
[tex]\[
27^2 \times 81^{0.5} = 3^6 \times 3^2
\][/tex]
Using the laws of exponents, specifically [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[
3^6 \times 3^2 = 3^{6+2} = 3^8
\][/tex]
Therefore, the expression [tex]\(27^2 \times 81^{0.5}\)[/tex] simplifies to [tex]\(3^8\)[/tex].
So, the correct choice is:
B. [tex]\(3^8\)[/tex]
