Let's break down the solution into two parts: expressing 11664 as a product of its prime factors in index form and then finding its square root.
### Step 1: Prime Factorization of 11664
To express 11664 as a product of its prime factors in index form, we need to determine the prime factors and their respective powers.
The prime factorization of 11664 is:
11664 can be factored as [tex]\( 2^4 \times 3^6 \)[/tex].
So, in index form, the factorization of 11664 is:
[tex]\[ 11664 = 2^4 \times 3^6 \][/tex]
### Step 2: Finding the Square Root of 11664
Next, we need to find the square root of 11664. We can use the property of exponents to do this.
The square root of a product can be found by taking the square root of each factor:
[tex]\[ \sqrt{2^4 \times 3^6} \][/tex]
Recall that:
[tex]\[ \sqrt{a^m} = a^{m/2} \][/tex]
So, applying this to each prime factor:
[tex]\[ \sqrt{2^4} = 2^{4/2} = 2^2 = 4 \][/tex]
[tex]\[ \sqrt{3^6} = 3^{6/2} = 3^3 = 27 \][/tex]
Now, multiply these results:
[tex]\[ \sqrt{11664} = 4 \times 27 = 108 \][/tex]
### Conclusion
The factorization of 11664 in index form is [tex]\( 2^4 \times 3^6 \)[/tex].
The square root of 11664 is [tex]\( \sqrt{11664} = 108 \)[/tex].
