What is the square root of 52 simplified?
The Correct Answer and Explanation is:
To simplify the square root of 52, we first break down the number into its prime factors and then simplify the square root expression.
Step 1: Factorization of 52
The first step is to find the prime factorization of 52. We divide 52 by the smallest prime numbers:
[
52 \div 2 = 26 \quad \text{(since 52 is even, we divide by 2)}
]
[
26 \div 2 = 13 \quad \text{(26 is also even, so divide by 2 again)}
]
[
13 \div 13 = 1 \quad \text{(13 is a prime number, so we stop here)}
]
Thus, the prime factorization of 52 is:
[
52 = 2 \times 2 \times 13 = 2^2 \times 13
]
Step 2: Simplifying the Square Root
Now, we take the square root of 52. We can express the square root as follows:
[
\sqrt{52} = \sqrt{2^2 \times 13}
]
Using the property of square roots, (\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}), we separate the square root into two parts:
[
\sqrt{52} = \sqrt{2^2} \times \sqrt{13}
]
Since the square root of (2^2) is 2, we get:
[
\sqrt{52} = 2 \times \sqrt{13}
]
Step 3: Final Answer
Thus, the simplified form of the square root of 52 is:
[
\sqrt{52} = 2\sqrt{13}
]
This is the simplest form of the square root of 52. We cannot simplify (\sqrt{13}) further, as 13 is a prime number and does not have a square root that simplifies into an integer.
Explanation:
The process of simplifying square roots involves finding the prime factors of the number under the root, grouping factors into pairs (because the square root of a pair of identical factors simplifies to a whole number), and pulling these out of the square root. For 52, we recognized that (2^2) forms a pair, allowing us to take 2 out of the square root, leaving (\sqrt{13}), which cannot be simplified further. This method ensures that the square root is expressed in its simplest form.