Let's solve the given expression step by step.
The expression provided is:
[tex]\[ (21)^{3/2} \times (21)^{5/2} \][/tex]
First, we'll use the property of exponents which states:
[tex]\[ a^m \times a^n = a^{m+n} \][/tex]
In this expression:
[tex]\[ a = 21 \][/tex]
[tex]\[ m = \frac{3}{2} \][/tex]
[tex]\[ n = \frac{5}{2} \][/tex]
Let's add the exponents:
[tex]\[ m + n = \frac{3}{2} + \frac{5}{2} \][/tex]
Since the denominators are the same, we can add the numerators:
[tex]\[ m + n = \frac{3 + 5}{2} = \frac{8}{2} = 4 \][/tex]
So, the expression simplifies to:
[tex]\[ 21^{4} \][/tex]
Next, we can calculate the power:
[tex]\[ 21^{4} \][/tex]
This can be broken down as:
[tex]\[ 21 \times 21 \times 21 \times 21 \][/tex]
Performing the multiplication step-by-step:
[tex]\[ 21 \times 21 = 441 \][/tex]
[tex]\[ 441 \times 21 = 9261 \][/tex]
[tex]\[ 9261 \times 21 = 194481 \][/tex]
Thus, the final result is:
[tex]\[ 21^4 = 194481 \][/tex]
Therefore, the value of \((21)^{3 / 2} \times (21)^{5 / 2}\) is \(\boxed{194481}\).
Additionally, calculating the intermediate terms:
[tex]\[ (21)^{3/2} \approx 96.234 \][/tex]
[tex]\[ (21)^{5/2} \approx 2020.916 \][/tex]
Multiplying these values:
[tex]\[ 96.234 \times 2020.916 \approx 194481.0 \][/tex]
So, the terms and final result confirm our entire calculation process.
