Question

Which of the following is a simpler form of the radical expression (27x^4) / (75y^2)?

Option 1: (3x^2) / (5y)
Option 2: (9x^2) / (15y^2)
Option 3: (3x^2) / (5y^3)
Option 4: (9x^4) / (75y^2)

Answer 1

The correct answer is option 1: the radical expression √(27x^4) / √(75y^2) simplifies to (3x^2) / (5y), achieved by taking the square root of both numerator and denominator and simplifying them independently.

Step-by-step explanation:

The correct answer is option 1: √(27x^4) / √(75y^2) simplifies to (3x^2) / (5y). To find this simpler form, we can take the square root of both the numerator and the denominator. The square root of 27 is 3³ which simplifies to 3, and x^4 is the square of x^2. So, √(27x^4) simplifies to 3x^2. In the denominator, the square root of 75 can be simplified to 5³y², which is 5 when y² is taken out of the square root, so √(75y^2) simplifies to 5y. Therefore, the fraction becomes (3x^2) / (5y).

To simplify the given radical expression, we need to find the prime factors of the numerator and denominator. The prime factors of 27x^4 are 3, 3, 3, x, x, x, x. And the prime factors of 75y^2 are 3, 5, 5, y, y.

Cancelling out the common factors, we are left with (3x^2) / (5y), which is equivalent to option 1.