simplify the complex fraction x/x+3/1/x+1/x+3
To simplify this complex fraction, we need to find a common denominator for the two fractions in the numerator and multiply:
x/(x+3) + 1/(x+3) = (x + 1)/(x+3)
Then, we can rewrite the denominator as a single fraction with a common denominator:
1/x + 1/(x+3) = (x+3+x)/(x(x+3)) = (2x+3)/(x^2+3x)
Putting it all together:
x / ((x+3)/(x+1) + (1/x + 1/(x+3)))
= x / ((x + 1)/(x+3) + (2x+3)/(x^2+3x))
= x / (((x+1)(x^2+3x)/(x+3)) + ((2x+3)(x+1)/(x+3)(x^2+3x)))
= x / ((x^3 + 5x^2 + 4x + 3)/(x^3 + 3x^2))
= x^4 + 3x^3 / (x^3 + 5x^2 + 4x + 3)
whats the final answer
The final answer is:
x^4 + 3x^3 / (x^3 + 5x^2 + 4x + 3)
To simplify the given complex fraction, we need to simplify the individual fractions within it and then combine them.
The complex fraction is:
x/x + 3/1/x + 1/x + 3
To simplify x/x, we can see that anything divided by itself is always equal to 1. So, x/x simplifies to 1.
The complex fraction becomes:
1 + 3/1/x + 1/x + 3
Next, let's simplify 3/1. Since any number divided by 1 is equal to itself, 3/1 simplifies to 3.
The complex fraction becomes:
1 + 3/x + 1/x + 3
Now, we need to find a common denominator for the two fractions 1/x and 1/x + 3. The common denominator is x(x + 3).
Multiplying the first fraction, 1/x, by (x + 3)/(x + 3) gives us (x + 3)/[x(x + 3)], while multiplying the second fraction, 1/x + 3, by x/x gives us x/[x(x + 3)].
The complex fraction becomes:
1 + (x + 3)/[x(x + 3)] + x/[x(x + 3)] + 3
Next, let's combine the fractions with a common denominator:
1 + (x + 3 + x)/[x(x + 3)] + 3
Simplifying the numerator:
1 + (2x + 3)/[x(x + 3)] + 3
Finally, combining the terms:
(1 + 3) + (2x + 3)/[x(x + 3)]
4 + (2x + 3)/[x(x + 3)]
Thus, the simplified form of the complex fraction x/x + 3/1/x + 1/x + 3 is 4 + (2x + 3)/[x(x + 3)].