What is the square root of 4356?

Steps to Solve

1. Identify the problem The goal is to find the square root of 4356, which means we need to find a number $x$ such that $x^2 = 4356$.

2. Use prime factorization or estimation We can either estimate the square root or use prime factorization. Let's try prime factorization.

3. Find prime factors of 4356 To find the prime factors, we divide 4356 by the smallest prime number and continue the process.

• Start by dividing 4356 by 2:
  $$ 4356 \div 2 = 2178 $$
• Divide 2178 by 2:
  $$ 2178 \div 2 = 1089 $$
• Now divide 1089 by 3 (the next smallest prime number):
  $$ 1089 \div 3 = 363 $$
• Divide 363 by 3:
  $$ 363 \div 3 = 121 $$
• Finally, divide 121 by 11:
  $$ 121 \div 11 = 11 $$
• And one last time:
  $$ 11 \div 11 = 1 $$

Thus, the full prime factorization of 4356 is $2^2 \times 3^2 \times 11^2$.

4. Square root from prime factors To find the square root, we take the square root of each prime factor: $$ \sqrt{4356} = \sqrt{2^2 \times 3^2 \times 11^2} = 2 \times 3 \times 11 $$

5. Calculate the final result Now multiply these values: $$ 2 \times 3 = 6 $$ $$ 6 \times 11 = 66 $$

Thus, the square root of 4356 is 66.