Express the square root 75 in
simplest radical form.
The Correct Answer and Explanation is :
The correct answer is: (5\sqrt{3})
To express the square root of 75 in simplest radical form, we start by factoring 75 into its prime factors. The prime factorization of 75 is:
[
75 = 3 \times 5^2
]
Next, we apply the property of square roots that states:
[
\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}
]
Using this property, we can rewrite (\sqrt{75}) as follows:
[
\sqrt{75} = \sqrt{3 \times 5^2}
]
Now, we can separate the factors under the square root:
[
\sqrt{75} = \sqrt{3} \times \sqrt{5^2}
]
Since (\sqrt{5^2}) equals 5, we can simplify further:
[
\sqrt{75} = \sqrt{3} \times 5
]
This gives us:
[
\sqrt{75} = 5\sqrt{3}
]
Thus, the simplest radical form of (\sqrt{75}) is (5\sqrt{3}).
Explanation
The process of simplifying square roots often involves breaking down the number into its prime factors. This is essential because square roots are only straightforward when the number under the radical can be expressed as a product of perfect squares. In our case, (5^2) is a perfect square, which simplifies nicely.
When simplifying radicals, it’s crucial to identify and extract the largest possible perfect square. In the case of 75, recognizing (5^2) allowed us to simplify the radical efficiently.
It’s also important to note that square roots cannot be simplified beyond their radical forms unless the number is a perfect square (e.g., (\sqrt{16} = 4)). Therefore, the resulting expression (5\sqrt{3}) is the simplest form, as (3) is not a perfect square.
In conclusion, the square root of 75 simplifies to (5\sqrt{3}) through prime factorization and applying properties of square roots. This method not only helps in solving mathematical problems but also enhances our understanding of the properties of numbers and their relationships.