What is the square root of 512?
Understand the Problem
The question is asking for the square root of the number 512. This requires a step-by-step calculation to determine the square root value.
Answer
16 \sqrt{2}
Answer for screen readers
The final answer is $16 \sqrt{2}$
Steps to Solve
- Break down the number into prime factors
First, find the prime factorization of 512.
$$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$$
Simplified, this is:
$$512 = 2^9$$
- Use the property of square roots and exponents
We know that the square root of a number can be expressed in terms of its prime factors:
$$\sqrt{a} = \sqrt{b^2} = b$$
So we have:
$$\sqrt{512} = \sqrt{2^9} = \sqrt{(2^4)^2 \times 2}$$
- Simplify the square root
We can take the square root of the term inside the parentheses and keep the square root of the remaining factor:
$$\sqrt{2^9} = 2^4 \times \sqrt{2} = 16 \sqrt{2}$$
The final answer is $16 \sqrt{2}$
More Information
The square root of 512 simplifies to $16 \sqrt{2}$, which shows a combination of an integer part and a simplest radical form.
Tips
A common mistake is to miscalculate the prime factors or to forget to simplify the square root properly. Always double-check your factorization and simplification steps.
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