Fully simplify.
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To fully simplify the expression:
1. Simplify the numerator of the first fraction: x - x^2
2. Simplify the denominator of the first fraction: x/3
3. Simplify the numerator of the second fraction: 3
4. Simplify the denominator of the second fraction: 3/x
The expression then becomes:
(x - x^2)/3 ÷ x/3 - 3/(x)
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
(x - x^2)/3 * 3/x - 3/(x)
Simplifying further:
(x - x^2)/x - 3/(x)
Now, to combine the fractions, we need a common denominator. The common denominator is x:
(x - x^2)/x + (-3x)/x
Combine the numerators over the shared denominator:
(x - x^2 - 3x)/x
Finally, we simplify the numerator by combining like terms:
(-x^2 - 2x)/x
This is the fully simplified expression.
To simplify the expression, we need to simplify each fraction first.
The first fraction is x - (x^2/3).
The second fraction is (x/3) - (3/x).
We can simplify the first fraction by multiplying both numerator and denominator by 3. This gives us 3x - x^2.
We can simplify the second fraction by finding a common denominator. The common denominator is 3x. So, the expression becomes [(x*x - 9)/3x].
Now, we can combine the two fractions and divide them by 3.
[(3x - x^2)/3] / [(x^2 - 9)/3x] can be simplified as (3x - x^2) / (x^2 - 9) * (3x/3).
Cancel out the 3s and we get (x - x^2) / (x^2 - 9) * (x/x).
Multiply (x - x^2) * (x) and (x^2 - 9) * (x) to get x(x - x^2) / (x - 3)(x + 3) * x.
Simplifying further, we get (x^2 - x^3) / (x^2 - 9) * x.
Therefore, the fully simplified expression is (x^2 - x^3) / (x^2 - 9) * x.
To fully simplify the expression, let's break it down step by step.
Step 1: Simplify the numerator.
The numerator consists of two fractions subtracted from each other. To simplify this, we need a common denominator. The common denominator in this case is 3x. So, we can rewrite the numerator as follows:
(x - x^2) / 3 = (x(1 - x)) / 3
Step 2: Simplify the denominator.
The denominator is already in its simplest form, so we don't need to do anything here.
Step 3: Divide the numerator by the denominator.
To divide by a fraction, we multiply by its reciprocal. The reciprocal of 3 is 1/3, so we have:
[(x(1 - x)) / 3] / (3/x) = (x(1 - x)) / 3 * (x/3)
Step 4: Simplify the expression further.
To simplify the expression, we can cancel out the common factors. In this case, we can cancel out one factor of x and 3:
(x(1 - x)) / 3 * (x/3) = (1 - x) / 1 = 1 - x
So, the fully simplified expression is 1 - x.