what is the simplified radical of the square root of 108.....use 2 and 54 as factors
thank you
2 * 54 = 2 * 2 * 27 = 2 * 2 * 3 * 9
√108 = 2 * 3 * √3 = 6√3
To find the simplified radical form of the square root of 108, we can start by factoring the number 108 into its prime factors.
Step 1: Prime factorization of 108
We can start by dividing 108 by the smallest prime number, which is 2.
108 ÷ 2 = 54
Now, we repeat the process by dividing 54 by 2.
54 ÷ 2 = 27
Since 27 is an odd number, we move on to the next prime number, which is 3.
27 ÷ 3 = 9
9 ÷ 3 = 3
Finally, we have factored 108 as:
108 = 2 × 2 × 3 × 3 × 3 = 2^2 × 3^3
Step 2: Expressing the square root using the factors
Now, let's express the square root of 108 in terms of its factors:
√108 = √(2^2 × 3^3)
Using the properties of square roots,
√(a × b) = √a × √b
We can rewrite the expression as:
√108 = √(2^2 × 3^3) = √(2^2) × √(3^3) = 2√3^3
Finally, we simplify further:
2√3^3 = 2 × 3√3 = 6√3
Therefore, the simplified radical form of the square root of 108, using factors 2 and 54, is 6√3.