What is square root 30 in simplified form?
The Correct answer and Explanation is:
The square root of 30, denoted as (\sqrt{30}), is not a perfect square, meaning it cannot be simplified into an integer or a simpler radical. However, it can be approximated or expressed in terms of its prime factors.
Understanding the Simplification Process
To simplify a square root, we generally look for factors of the number under the radical that are perfect squares. A perfect square is a number that can be expressed as the product of an integer multiplied by itself. The perfect squares less than 30 are 1, 4 (which is (2^2)), 9 (which is (3^2)), and 16 (which is (4^2)), and 25 (which is (5^2)).
Prime Factorization of 30
The first step in examining (\sqrt{30}) is to perform its prime factorization. The number 30 can be expressed as the product of its prime factors:
[
30 = 2 \times 3 \times 5
]
Since none of these factors is a perfect square, (\sqrt{30}) cannot be simplified further. Thus, (\sqrt{30}) remains in its simplest radical form.
Decimal Approximation
For practical purposes, (\sqrt{30}) can also be approximated as a decimal. Using a calculator, we find:
[
\sqrt{30} \approx 5.477
]
This approximation indicates that the square root of 30 is slightly less than 5.5.
Conclusion
In summary, the square root of 30 in its simplified form is (\sqrt{30}) itself. It does not simplify to a cleaner radical or integer due to the absence of perfect square factors. Its approximate decimal value is about 5.477. Understanding square roots and their simplifications is fundamental in mathematics, particularly in algebra, where they often arise in solving equations and modeling real-world problems.