Radical and Rational Exponent

find roots
square roots of 12a^3/25=6a^3

-3-square roots 18/-6=-1

check this for me it find the roots of the problem.

To find the roots of the equation, you need to solve for the value of "a" that makes the equation true. Let's break down the given equation step by step:

1. Start with the equation: √(12a^3/25) = 6a^3

2. Square both sides of the equation to get rid of the square root: (√(12a^3/25))^2 = (6a^3)^2

3. Simplify: 12a^3/25 = 36a^6

4. Multiply both sides of the equation by 25 to get rid of the fraction: 12a^3 = 900a^6

5. Rearrange the equation: 900a^6 - 12a^3 = 0

Now, we have a polynomial equation. To find the roots of this equation, we can use factoring:

6. Factor out the common term "12a^3" from both terms: 12a^3(75a^3 - 1) = 0

7. Set each factor equal to zero and solve for 'a':

12a^3 = 0 ---> a = 0

75a^3 - 1 = 0

8. Solve for 'a' by isolating the variable:

75a^3 = 1

Divide both sides of the equation by 75: a^3 = 1/75

Take the cube root of both sides: a = ∛(1/75)

Therefore, the roots of the equation are a = 0 and a = ∛(1/75).

However, in the last step, we cannot simplify the cube root any further without using decimal approximations. So, it is best to write the solution as a decimal or in radical form, depending on the requirements of the problem.