To solve for [tex]\( n \)[/tex] in the equation [tex]\( 3 \times \sqrt{27} = 3^n \)[/tex], we can follow these steps:
1. Simplify the square root of 27:
[tex]\[
\sqrt{27} = 27^{1/2}
\][/tex]
2. Express 27 in terms of its prime factors:
[tex]\[
27 = 3^3
\][/tex]
Therefore:
[tex]\[
27^{1/2} = (3^3)^{1/2}
\][/tex]
3. Apply the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(3^3)^{1/2} = 3^{3 \cdot \frac{1}{2}} = 3^{3/2}
\][/tex]
4. Substitute back into the original equation:
[tex]\[
3 \times \sqrt{27} = 3 \times 3^{3/2}
\][/tex]
5. Simplify using the properties of exponents [tex]\(a \times a^b = a^{1 + b}\)[/tex]:
[tex]\[
3 \times 3^{3/2} = 3^{1 + 3/2} = 3^{2.5} = 3^{5/2}
\][/tex]
6. Equate the exponents:
[tex]\[
3^n = 3^{5/2}
\][/tex]
Therefore:
[tex]\[
n = \frac{5}{2} = 2.5
\][/tex]
So, the value of [tex]\( n \)[/tex] is [tex]\( \frac{5}{2} \)[/tex] or [tex]\( 2.5 \)[/tex].
