Calculating the Square Root of 576

The question asks for the value that, when multiplied by itself, equals 576. This is known as finding the square root of 576, denoted mathematically as $\sqrt{576}$.

Method: Prime Factorization

We can find the square root by breaking down 576 into its prime factors:

• Start dividing 576 by the smallest prime number, 2:
  • $576 \div 2 = 288$
  • $288 \div 2 = 144$
  • $144 \div 2 = 72$
  • $72 \div 2 = 36$
  • $36 \div 2 = 18$
  • $18 \div 2 = 9$
• Now, divide by the next prime number, 3:
  • $9 \div 3 = 3$
  • $3 \div 3 = 1$

The prime factorization of 576 is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3$, or $2^6 \times 3^2$.

Finding the Square Root Value

To find the square root, we take half of the exponent for each prime factor:

• $\sqrt{576} = \sqrt{2^6 \times 3^2}$
• $= 2^{6/2} \times 3^{2/2}$
• $= 2^3 \times 3^1$
• $= (2 \times 2 \times 2) \times 3$
• $= 8 \times 3$
• $= 24$

Therefore, the square root of 576 is 24.

Final Answer: The final answer is 24