simplify the complex fraction
x/(x+2)/((1/x)+1/(x+2))
if you put that in the calculator it should show you what it looks like properly.
A: x^2/2
B: x^2/x+2
C: x^2/2x+2
D: x^2+2x+2/2x+2
I think it is C
correct
Well, my dear friend, let's put on our math clown noses and figure this out together!
To simplify the complex fraction, we need to find the least common denominator (LCD) of the denominator. In this case, the LCD is (x)(x+2).
To get rid of the fraction in the numerator, we can multiply both the numerator and denominator by the LCD:
x/(x+2)/((1/x) + 1/(x+2)) * (x)(x+2)/(x)(x+2)
Now, let's simplify the numerator by distributing the LCD:
x * (x)(x+2) = x(x^2 + 2x) = x^3 + 2x^2
And the denominator becomes:
(x)(x+2) * (1/x) + (x)(x+2) * (1/(x+2)) = (x+2) + x = 2x + 2
Putting it all together, the complex fraction simplifies to:
(x^3 + 2x^2)/(2x + 2)
So, after all the clown calculations, the correct answer is (D) x^2 + 2x + 2 / 2x + 2.
Hope this math circus act made you smile!
To simplify the given complex fraction:
x/(x+2)/((1/x)+1/(x+2)),
we can follow these steps.
Step 1: Simplify the numerator of the complex fraction by multiplying the terms using the distributive property:
x * (1/x) = 1,
x * (1/(x+2)) = 1/(x+2).
Now the numerator becomes 1 + 1/(x+2).
Step 2: Simplify the denominator of the complex fraction by finding a common denominator for the terms (1/x) and (1/(x+2)):
Common denominator = x(x+2).
(1/x) * (x+2)/(x+2) = (x+2)/(x(x+2)),
(1/(x+2)) * x/x = x/x(x+2).
Now the denominator becomes (x+2)/(x(x+2)) + x/x(x+2).
Step 3: Combine the simplified numerator and denominator:
(1 + 1/(x+2)) / ((x+2)/(x(x+2)) + x/x(x+2)).
To simplify further, we can multiply the numerator and denominator by the reciprocal of the denominator:
[(1 + 1/(x+2)) * (x(x+2))] / [(x+2)/(x(x+2)) + x/x(x+2)]
Simplifying:
[(x(x+2) + x(x+2)/(x+2)] / [(x+2) + x]
Simplifying further:
[(x^2 + 2x + x^2 + 2x)/(x+2)] / [(x+2) + x]
= (2x^2 + 4x) / (2x + 2)
= 2x(x + 2) / 2(x + 1)
= x(x + 2) / (x + 1)
Thus, the simplified form of the given complex fraction is x(x + 2) / (x + 1).
Comparing this expression with the options given, the correct answer is B: x^2/(x+1).
To simplify the given complex fraction, let's start by simplifying the numerator and denominator separately.
The numerator is x.
To simplify the denominator, we need to combine the fractions [(1/x) and (1/(x+2)) into a single fraction.
To do that, we need to find a common denominator, which is (x)(x+2), for both fractions.
The first fraction (1/x) can be multiplied by (x+2)/(x+2) to get (x+2)/(x(x+2)).
The second fraction (1/(x+2)) can be multiplied by (x)/(x) to get (x)/(x(x+2)).
Now we can combine both fractions into a single fraction: (x+2 + x) / (x(x+2)) = (2x+2) / (x(x+2)).
Now our complex fraction becomes: x / ((2x+2) / (x(x+2))).
To divide by a fraction, we can multiply by its reciprocal. In this case, we can multiply the complex fraction by (x(x+2))/(2x+2).
(x / 1) * (x(x+2) / (2x+2)) = (x * x(x+2)) / (2x+2)
Simplifying the numerator:
x * x(x+2) = x^2(x+2) = x^3 + 2x^2
Simplifying the denominator:
2x+2 = 2(x+1)
Putting it all together:
(x^3 + 2x^2) / (2(x+1))
So the simplified expression of the complex fraction is (x^3 + 2x^2) / (2(x+1)).
Comparing this with the given options, we can see that the answer is not option C: x^2/2x+2.
Looking at the options:
A: x^2/2
B: x^2/x+2
D: (x^2+2x+2)/(2x+2)
We can see that the simplified expression (x^3 + 2x^2) / (2(x+1)) does not match any of the given options. It seems that none of the options provided is correct.