Express in simplest radical form.
square root of, 5, end square root, square root of, 30, end square root
5
30
√5 * √30
Simplifying it further, we can extract the common factors from the square roots:
√(5 * 30)
√(150)
The prime factorization of 150 is 2 * 3 * 5 * 5. Since there are two 5's, one can be taken out of the square root:
5√6
So the expression simplified to simplest radical form is 5√6.
To express the square root of 5 over the square root of 30 in simplest radical form, we can simplify the expression by rationalizing the denominator.
First, let's write the expression:
√5 / √30
To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator.
The conjugate of √30 is -√30 (the negative square root of 30).
So, multiplying the numerator and the denominator by -√30:
(√5 / √30) * (-√30 / -√30)
This simplifies to:
-√(5*30) / √(30*30)
Simplifying the square roots:
-√(150) / √(900)
Now, we divide the numbers inside the square roots:
-√150 / √900
The square root of 150 cannot be simplified further, but the square root of 900 can be simplified to 30:
-√150 / 30
Now, we can simplify further by dividing both the numerator and the denominator by their greatest common factor (GCF), which is 10:
- (1/10) * √150 / (1/10) * 30
Simplifying:
- √150 / 10
Now, we can simplify further by dividing the square root of 150 by 10:
- √(150/10)
Simplifying further:
- √15
So, the expression √5 / √30 simplifies to -√15 in simplest radical form.
To express the given expression in simplest radical form, we need to simplify the square roots.
Let's break down our given expression: √5 + √30.
To simplify √5, we need to check if 5 has any perfect square factors. Since 5 does not have any perfect square factors other than 1, the square root of 5 cannot be simplified further.
To simplify √30, we need to find the prime factorization of 30. It can be written as 2 × 3 × 5. Now, we look for perfect square factors. We see that 2 and 3 do not have perfect square factors, but 5 is a perfect square factor. We can simplify it by taking the square root of 5, which is just √5.
So, the simplified radical form of √5 + √30 is √5 + √(2 × 3 × 5).
Combine the like terms under the same square root: √5 + √(2 × 3 × 5) = √5 + √(6 × 5).
Since both terms have a common factor of 5 under the square root, we can factor it out: √5 + √(5 × 6) = √5 + √5 × √6.
The square root of 5 can be written as √5 × 1: √5 + 1 × √6.
Now, we can rewrite our expression as: √5(1 + √6).
So, the simplified radical form of √5 + √30 is √5(1 + √6).