Square root of 75 simplified
The Correct Answer and Explanation is :
The correct answer is: (5\sqrt{3})
To simplify the square root of 75, we break it down into its prime factors:
[
\sqrt{75} = \sqrt{25 \times 3}
]
Next, we use the fact that the square root of a product is the product of the square roots:
[
\sqrt{75} = \sqrt{25} \times \sqrt{3}
]
Since (\sqrt{25} = 5), we get:
[
\sqrt{75} = 5\sqrt{3}
]
Thus, the simplified form of (\sqrt{75}) is (5\sqrt{3}).
Explanation:
The goal of simplifying square roots is to express them in a form that contains the smallest possible radical. For (\sqrt{75}), we first need to break 75 into its prime factors. Prime factorization helps us identify perfect squares, which can be simplified.
The number 75 can be factored as:
[
75 = 25 \times 3
]
Here, 25 is a perfect square, and (\sqrt{25}) is 5. This allows us to simplify the square root of the product. The square root of 75 then becomes:
[
\sqrt{75} = 5\sqrt{3}
]
The square root of 3 remains under the radical because 3 is a prime number and cannot be simplified further. So, the expression (5\sqrt{3}) is the simplest form of (\sqrt{75}).
This process of simplifying square roots by factoring out perfect squares is a common technique in algebra. It makes calculations easier and expressions neater, especially when dealing with irrational numbers, like the square root of 3, which cannot be simplified into a whole number.