What is the square root of 1152

What is the square root of 1152?

The correct answer and explanation is:

To find the square root of 1152 using prime factorization, follow these steps:

1. Prime Factorization of 1152:
   Start by dividing 1152 by the smallest prime number (2), as 1152 is even.
   • 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 move to the next prime number, 3).
   • 9 ÷ 3 = 3
   • 3 ÷ 3 = 1
   Now, we have the prime factorization of 1152:
   1152=27×321152 = 2^7 \times 3^2
2. Square Root Calculation:
   To find the square root, take half the power of each prime factor in the prime factorization. This is because when we take the square root, we are essentially reversing the square of each prime factor.
   • For 272^7, take half of 7, which is 3.5. This gives us 23×20.52^3 \times 2^{0.5}, so we take 232^3 as part of the square root.
   • For 323^2, take half of 2, which is 1. This gives us 313^1.
   Thus, the square root of 1152 is: 1152=23×31×2\sqrt{1152} = 2^3 \times 3^1 \times \sqrt{2} Simplifying the factors: 1152=8×3×2=242\sqrt{1152} = 8 \times 3 \times \sqrt{2} = 24 \sqrt{2}
3. Approximate Value:
   Since 2≈1.414\sqrt{2} \approx 1.414, multiply this with 24: 24×1.414≈33.93624 \times 1.414 \approx 33.936 So, the square root of 1152 is approximately 33.94.