Perform the indicated operations and simplify.
x^2/3(square root of x-1/square root of x)
x^2/3 * (x^1/2 -1)/x^1/2
x^2/3 * (1 - x^-1/2)
x^2/3 - x^1/6
20 cos t=-8
To simplify the expression, we need to simplify each term separately and then combine them.
Let's start with the first term, x^2/3. This means we have x^2 raised to the power of 1/3.
To simplify this, we need to find the cube root of x^2. Cube root is the opposite operation of raising a number to the power of 3.
So, the cube root of x^2 is (x^2)^(1/3) = x^(2/3).
Next, let's move on to the second term, (square root of x - 1) / (square root of x).
We can simplify this by rationalizing the denominator, which means removing any radicals from the denominator.
To do that, we multiply both the numerator and denominator by the conjugate of the denominator, which is (sqrt(x) + 1).
((sqrt(x) - 1) / (sqrt(x))) * ((sqrt(x) + 1) / (sqrt(x) + 1))
Expanding the numerator, we get (sqrt(x) - 1)(sqrt(x) + 1) = x - 1.
Expanding the denominator, we get (sqrt(x))(sqrt(x) + 1) = x + sqrt(x).
So, the simplified form of the second term is (x - 1) / (x + sqrt(x)).
Now we can combine the simplified forms of each term:
(x^(2/3) * (x - 1)) / (x + sqrt(x))
This is the simplified expression.