3√(5a) - 8√(a) / 35a^2
To simplify this expression, we can first separate the two terms in the numerator:
3√(5a) / 35a^2 - 8√(a) / 35a^2
We can then simplify each term separately. For the first term:
3√(5a) / 35a^2 = √(5a) / (35a^2/3)
We can simplify the denominator by using the rule that (a^m)^n = a^(mn):
35a^2/3 = (5a^2/3) * 7 = (a^(2/3))^5 * 7
So the first term simplifies to:
√(5a) / (a^(2/3))^5 * 7
For the second term:
8√(a) / 35a^2 = 8a^(1/2) / (35a^2)
We can simplify the denominator by using the rule that a^-m = 1/a^m:
35a^2 = 5 * 7 * a^2 = a^-2 * 5 * 7
So the second term simplifies to:
8a^(1/2) / (a^-2 * 5 * 7)
Now we can combine the two terms by finding a common denominator:
√(5a) / (a^(2/3))^5 * 7 - 8a^(1/2) / (a^-2 * 5 * 7)
The common denominator is (a^(2/3))^5 * 5 * 7:
√(5a) * 5 * 7 / (a^(2/3))^5 * 5 * 7 - 8a^(1/2) * (a^(2/3))^5 * 5 * 7 / (a^-2 * 5 * 7) * (a^(2/3))^5 * 5 * 7
Simplifying each term, we get:
35√(5a) / 5a^2 - 40a^(5/2) / 5a^3
Now we can simplify further by factoring out a common factor of 5a^(5/2):
5a^(5/2) * (7√(5a) - 8) / 5a^3
The 5's cancel out, and we can remove the common factor of 5a^(5/2) to get:
(7√(5a) - 8) / a^(3/2)
So the simplified expression is:
(7√(5a) - 8) / a^(3/2)