Subtract:
F(x) = x^2+7x-8/x^3-5
G(x) = -7x-5x^3/4square root of x
looks pretty straightforward, if I read it right.
f(x)-g(x) = (x^2+7x-8)/(x^3-5) - (-7x-5x^3)/(4√x)
so, it looks like we need to use a common denominator, so that gives us
(x^2+7x-8)(4√x) + (5x^3+7x)(x^3-5)
-------------------------------------------------------------
(4√x)(x^3-5)
Now just expand all that out if you wish. That pesky √x does get in the way, though.
To subtract the given functions F(x) and G(x), we need to find a common denominator for the fractions. Let's start with each function separately.
For F(x):
F(x) = (x^2 + 7x - 8) / (x^3 - 5)
For G(x):
G(x) = (-7x - 5x^3) / (4√x)
To find a common denominator, we need to simplify G(x) further. Let's simplify the denominator first.
The denominator of G(x) can be rewritten as:
4√x = 4 * x^(1/2) = 4x^(1/2)
Now the expression for G(x) becomes:
G(x) = (-7x - 5x^3) / (4x^(1/2))
To find a common denominator, we multiply the numerator and denominator of F(x) by 4x^(1/2):
F(x) = [(x^2 + 7x - 8) * 4x^(1/2)] / [(x^3 - 5) * 4x^(1/2)]
= (4x^(3/2) + 28x^(3/2) - 32x^(1/2)) / (4x^(7/2) - 20x^(1/2))
Now, we can subtract the functions by subtracting their respective numerators and keeping the common denominator:
F(x) - G(x) = [(4x^(3/2) + 28x^(3/2) - 32x^(1/2)) - (-7x - 5x^3)] / (4x^(7/2) - 20x^(1/2))
Simplifying the numerator:
F(x) - G(x) = (4x^(3/2) + 28x^(3/2) - 32x^(1/2) + 7x + 5x^3) / (4x^(7/2) - 20x^(1/2))
And there you have it. The subtraction of the given functions F(x) and G(x) is given by:
F(x) - G(x) = (4x^(3/2) + 28x^(3/2) - 32x^(1/2) + 7x + 5x^3) / (4x^(7/2) - 20x^(1/2))