Simplify the radical expression. sqrt 30/120n^2
To simplify the given radical expression, we need to simplify both the numerator and denominator separately.
Let's start with the numerator, which is √30.
To simplify the square root of 30, we can factorize 30. The prime factorization of 30 is 2 * 3 * 5. Now, we can pair up the prime factors:
√30 = √(2 * 3 * 5)
Using the property of square roots, we can write it as a product of individual square roots:
√30 = √2 * √3 * √5
Now, the square roots of 2, 3, and 5 cannot be further simplified because they are prime numbers. So, we leave them as they are:
√30 = √2 * √3 * √5
Similarly, let's simplify the denominator, which is √(120n^2). First, let's factorize 120 to simplify the square root:
√(120n^2) = √(2 * 2 * 2 * 3 * 5 * n^2)
Since 2 and 5 have pairs, we can take them out of the square root:
√(120n^2) = √(2^2 * 2 * 3 * 5 * n^2)
Using the property of square roots again, we can simplify this expression:
√(120n^2) = 2 * n * √(2 * 3 * 5)
Now, let's simplify the whole expression by dividing the simplified numerator by the simplified denominator:
√30 / √(120n^2) = (√2 * √3 * √5) / (2 * n * √(2 * 3 * 5))
At this point, notice that we can cancel out the common factor of √5 on the numerator and the denominator:
√30 / √(120n^2) = (√2 * √3 * √5) / (2 * n * √(2 * 3 * 5))
= (√2 * √3) / (2 * n)
Finally, we simplify further by multiplying the square roots on the numerator:
√30 / √(120n^2) = (√2 * √3) / (2 * n)
= √6 / (2 * n)
Therefore, the simplified radical expression is √6 / (2n).