find the root of 1152 by using prime factorisation ?
The correct answer and explanation is:
To find the square root of 1152 using prime factorization, we start by breaking 1152 down into its prime factors.
Step 1: Prime Factorization of 1152
First, we divide 1152 by the smallest prime number, which is 2.
1152 ÷ 2 = 576
576 ÷ 2 = 288
288 ÷ 2 = 144
144 ÷ 2 = 72
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
Now, 9 is not divisible by 2, so we move to the next prime number, which is 3.
9 ÷ 3 = 3
3 ÷ 3 = 1
So, the prime factorization of 1152 is:
1152 = 2⁷ × 3²
Step 2: Taking the Square Root
To take the square root of a number using prime factorization, we use the property that the square root of a product is the product of the square roots of the factors.
The square root of a number is found by dividing the powers of the prime factors by 2, since we are looking for pairs of primes.
For 1152 = 2⁷ × 3², the square root is calculated as:
√1152 = √(2⁷ × 3²)
= 2^(7/2) × 3^(2/2)
= 2³ × 2^(1/2) × 3
= 8 × √2 × 3
Step 3: Approximation
Since √2 is approximately 1.414, the square root of 1152 is:
√1152 ≈ 8 × 1.414 × 3
≈ 8 × 4.242
≈ 33.936
Thus, the square root of 1152 is approximately 33.94.
Conclusion
The exact square root is 8√2, but an approximate value is 33.94. This method of prime factorization helps break down the number into its prime components, making it easier to handle square roots, especially with non-perfect squares.