Simplify square root of 105
The Correct Answer and Explanation is :
To simplify the square root of 105, we start by factoring the number into its prime factors. The prime factorization of 105 is:
[
105 = 3 \times 5 \times 7
]
Since none of these factors are perfect squares, the square root of 105 cannot be simplified further in terms of whole numbers. Therefore, we express the simplified form of (\sqrt{105}) as it is, but we can approximate its decimal value.
To find a decimal approximation, we can use a calculator:
[
\sqrt{105} \approx 10.247
]
Detailed Explanation:
- Understanding Square Roots:
The square root of a number (x) is a value (y) such that (y^2 = x). For example, (\sqrt{9} = 3) because (3^2 = 9). However, when (x) is not a perfect square (like 105), we can’t express (\sqrt{x}) as a whole number. - Prime Factorization:
To simplify a square root, we often use prime factorization. This method helps us identify perfect squares within the number. In the case of 105, breaking it down gives us (3), (5), and (7), all of which are prime and do not have pairs (as pairs lead to perfect squares). - Perfect Squares:
A perfect square is a number that can be expressed as the square of an integer (e.g., (1, 4, 9, 16,) etc.). When simplifying square roots, any factor that appears twice can be taken out of the square root. Since 105 has no repeated factors, it cannot be simplified further. - Conclusion:
Thus, (\sqrt{105}) remains as (\sqrt{105}) or can be approximated as (10.247). It’s important to understand that simplification mainly applies when we can extract perfect squares from under the square root. In summary, (\sqrt{105}) is best expressed in its simplest form as (\sqrt{105}) or approximately (10.247).