To solve for [tex]\( m \)[/tex] in the given equation [tex]\( 3 \times \sqrt{27} = 3^m \)[/tex], we need to simplify the left-hand side and express it in a form that allows us to compare it directly to the right-hand side.
### Step-by-Step Solution:
1. Simplify the left-hand side:
[tex]\[
3 \times \sqrt{27}
\][/tex]
First, express 27 as a power of 3:
[tex]\[
27 = 3^3
\][/tex]
Thus, the square root of 27 is:
[tex]\[
\sqrt{27} = \sqrt{3^3} = 3^{3/2}
\][/tex]
Substituting this back into the expression gives:
[tex]\[
3 \times \sqrt{27} = 3 \times 3^{3/2}
\][/tex]
2. Combine the terms:
[tex]\[
3 \times 3^{3/2}
\][/tex]
Using the property of exponents [tex]\( a^b \times a^c = a^{b+c} \)[/tex], we can combine the terms:
[tex]\[
3^1 \times 3^{3/2} = 3^{1 + 3/2}
\][/tex]
Simplify the exponent:
[tex]\[
1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2}
\][/tex]
So we have:
[tex]\[
3 \times \sqrt{27} = 3^{5/2}
\][/tex]
3. Compare to the right-hand side:
[tex]\[
3^{5/2} = 3^m
\][/tex]
Since the bases are equal, we can set the exponents equal to each other:
[tex]\[
m = \frac{5}{2}
\][/tex]
### Conclusion:
The value of [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{5}{2}
\][/tex]
