To simplify the expression [tex]\( 3^{\frac{1}{2}} \times 3^{\frac{3}{2}} \)[/tex], we can use the properties of exponents. Specifically, we will use the property that states:
[tex]\[ a^m \times a^n = a^{m+n} \][/tex]
Here, our base [tex]\( a \)[/tex] is 3, and the exponents [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{3}{2}\)[/tex], respectively.
1. Step 1: Add the exponents:
[tex]\[ \frac{1}{2} + \frac{3}{2} \][/tex]
Since the denominators are the same, we can directly add the numerators:
[tex]\[ \frac{1 + 3}{2} = \frac{4}{2} = 2 \][/tex]
2. Step 2: Simplify the expression using the summed exponent:
[tex]\[ 3^{\frac{1}{2} + \frac{3}{2}} = 3^2 \][/tex]
3. Step 3: Evaluate the power:
[tex]\[ 3^2 = 9 \][/tex]
So, the simplified form of [tex]\( 3^{\frac{1}{2}} \times 3^{\frac{3}{2}} \)[/tex] is [tex]\( 9 \)[/tex].
Therefore, the final answer is:
[tex]\[ 3^{\frac{1}{2}} \times 3^{\frac{3}{2}} = 9 \][/tex]
