Question

the movement of the progress bar may be uneven because questions can be worth more or less (incl
simplify: (\frac{\frac{2}{x}+\frac{1}{3x^2}}{1+\frac{1}{6x}})
(\bigcirc) (\frac{2}{x})
(\bigcirc) (\frac{12x + 2}{6x^2 + x})
(\bigcirc) (2x)
(\bigcirc) (\frac{4}{x})

Explanation:

Step1: Simplify the numerator of the complex fraction

The numerator is $\frac{2}{x} + \frac{1}{3x^2}$. Find a common denominator, which is $3x^2$.
$\frac{2}{x} = \frac{2\times3x}{x\times3x} = \frac{6x}{3x^2}$
So, $\frac{2}{x} + \frac{1}{3x^2} = \frac{6x}{3x^2} + \frac{1}{3x^2} = \frac{6x + 1}{3x^2}$

Step2: Simplify the denominator of the complex fraction

The denominator is $1 + \frac{1}{6x}$. Find a common denominator, which is $6x$.
$1 = \frac{6x}{6x}$
So, $1 + \frac{1}{6x} = \frac{6x}{6x} + \frac{1}{6x} = \frac{6x + 1}{6x}$

Step3: Divide the simplified numerator by the simplified denominator

We have the complex fraction $\frac{\frac{6x + 1}{3x^2}}{\frac{6x + 1}{6x}}$. Dividing by a fraction is multiplying by its reciprocal:
$\frac{6x + 1}{3x^2} \times \frac{6x}{6x + 1}$

Step4: Cancel out common factors

The $(6x + 1)$ terms cancel out, and we can simplify the $x$ and constant terms:
$\frac{6x}{3x^2} = \frac{6}{3x} = \frac{2}{x}$ (Wait, no, wait. Wait, 6x divided by 3x²: 6/3 is 2, x/x² is 1/x, so 2/x? Wait, no, let's check again. Wait, 6x in numerator, 3x² in denominator. So 6/3 is 2, x/x² is 1/x, so 2/x? But wait, let's do the multiplication again. Wait, $\frac{6x + 1}{3x^2} \times \frac{6x}{6x + 1}$: the (6x + 1) cancels, so we have (6x)/(3x²) = (6)/(3x) = 2/x? Wait, but that's one of the options, but let's check the options again. Wait, no, maybe I made a mistake. Wait, wait, the numerator after step1: $\frac{2}{x} + \frac{1}{3x^2}$: common denominator 3x², so 2*3x/(3x²) + 1/(3x²) = (6x + 1)/(3x²). Denominator: 1 + 1/(6x) = (6x + 1)/(6x). Then the complex fraction is [(6x + 1)/(3x²)] / [(6x + 1)/(6x)] = [(6x + 1)/(3x²)] * [6x/(6x + 1)] = (6x)/(3x²) = 2/x. Wait, but option B is (12x + 2)/(6x² + x). Wait, maybe my initial simplification of the numerator was wrong. Wait, let's re-express the original numerator: $\frac{2}{x} + \frac{1}{3x^2}$. Let's get a common denominator of 3x²: 2/x = 6x/(3x²), so 6x/(3x²) + 1/(3x²) = (6x + 1)/(3x²). Denominator: 1 + 1/(6x) = (6x + 1)/(6x). Then the fraction is [(6x + 1)/(3x²)] / [(6x + 1)/(6x)] = (6x + 1)/(3x²) * 6x/(6x + 1) = 6x/(3x²) = 2/x. But option B is (12x + 2)/(6x² + x). Let's check if that's equivalent to 2/x. Let's factor numerator and denominator of B: numerator 12x + 2 = 2(6x + 1), denominator 6x² + x = x(6x + 1). So (2(6x + 1))/(x(6x + 1)) = 2/x. Oh! So option B simplifies to 2/x? Wait, no, wait: (12x + 2)/(6x² + x) = 2(6x + 1)/(x(6x + 1)) = 2/x. So both the result we got and option B are equivalent? Wait, but let's check the options again. Wait, the options are:

A. 2/x

B. (12x + 2)/(6x² + x)

C. 2x

D. 4/x

Wait, so when we simplified, we got 2/x, which is option A. But option B, when simplified, also gives 2/x. Wait, maybe I made a mistake in the initial steps. Wait, let's re-express the original problem:

The original expression is $\frac{\frac{2}{x} + \frac{1}{3x^2}}{1 + \frac{1}{6x}}$

Let's multiply numerator and denominator by 6x² to eliminate all denominators.

Numerator: 6x²*($\frac{2}{x} + \frac{1}{3x^2}$) = 6x²*(2/x) + 6x²*(1/3x²) = 12x + 2

Denominator: 6x²*(1 + $\frac{1}{6x}$) = 6x²*1 + 6x²*(1/6x) = 6x² + x

So the expression becomes (12x + 2)/(6x² + x), which is option B. And when we simplify option B, we get 2/x, but the question is to simplify the original expression, and option B is a simplified form (maybe not the most simplified, but let's check the options). Wait, but when we did the first method, we got 2/x, which is option A. But there's a contradiction here. Wait, no, let's check the multiplication by 6x². Let's do that:

Multiply numerator and denominator by 6x² (the least common multiple of x, 3x², 6x, which is 6x²):

Numerator: 6x²*($\frac{2}{x}$) + 6x²*($\frac{1}{3x^2}$) = 12x + 2

Denominator: 6x²*1 + 6x²*($\frac{1}{6x}$) = 6x² + x

So the expression is (12x + 2)/(6x² + x), which is option B. Now, let's factor numerator and denominator:

Numerator: 2(6x + 1)

Denominator: x(6x + 1)

So we can cancel (6x + 1) (assuming 6x + 1 ≠ 0, i.e., x ≠ -1/6) to get 2/x, which is option A. But the options include both A and B. Wait, maybe the problem considers (12x + 2)/(6x² + x) as a simplified form, or maybe I made a mistake in the first method. Wait, let's check the original problem again.

Wait, the original numerator is $\frac{2}{x} + \frac{1}{3x^2}$, denominator is $1 + \frac{1}{6x}$.

First, simplify numerator:

$\frac{2}{x} + \frac{1}{3x^2} = \frac{6x + 1}{3x^2}$ (common denominator 3x²)

Denominator:

$1 + \frac{1}{6x} = \frac{6x + 1}{6x}$ (common denominator 6x)

So the complex fraction is $\frac{\frac{6x + 1}{3x^2}}{\frac{6x + 1}{6x}} = \frac{6x + 1}{3x^2} \times \frac{6x}{6x + 1} = \frac{6x}{3x^2} = \frac{2}{x}$

But when we multiply numerator and denominator by 6x², we get (12x + 2)/(6x² + x), which factors to 2(6x + 1)/(x(6x + 1)) = 2/x. So both A and B are equivalent, but maybe the problem expects the form in option B? Wait, no, let's check the options again. Wait, the options are:

A. 2/x

B. (12x + 2)/(6x² + x)

C. 2x

D. 4/x

Wait, when we multiply numerator and denominator by 6x², we get (12x + 2)/(6x² + x), which is option B. And when we simplify that, we get 2/x, which is option A. But maybe the problem considers B as a simplified form before canceling the (6x + 1) term. Wait, but let's check the arithmetic again. Wait, 6x²*(2/x) is 12x, and 6x²*(1/3x²) is 2, so numerator is 12x + 2. Denominator: 6x²*1 is 6x², and 6x²*(1/6x) is x, so denominator is 6x² + x. So the expression becomes (12x + 2)/(6x² + x), which is option B. Then, factoring, we get 2(6x + 1)/(x(6x + 1)) = 2/x, which is option A. But maybe the problem has a typo, or maybe I made a mistake. Wait, let's check with x = 1. Let's plug x = 1 into the original expression:

Original expression: numerator: 2/1 + 1/(3*1²) = 2 + 1/3 = 7/3; denominator: 1 + 1/(6*1) = 7/6; so the expression is (7/3)/(7/6) = (7/3)*(6/7) = 2. Now check option A: 2/1 = 2. Option B: (12*1 + 2)/(6*1² + 1) = 14/7 = 2. Oh! So both A and B give 2 when x = 1. Let's try x = 2. Original expression: numerator: 2/2 + 1/(3*4) = 1 + 1/12 = 13/12; denominator: 1 + 1/(12) = 13/12; so expression is (13/12)/(13/12) = 1. Option A: 2/2 = 1. Option B: (24 + 2)/(24 + 2) = 26/26 = 1. So both A and B are equivalent. But the options include both. Wait, but maybe the problem expects the form in option B? Wait, no, the question is to simplify, and 2/x is simpler than (12x + 2)/(6x² + x). But maybe the problem's options have a mistake, or maybe I made a mistake. Wait, let's check the initial steps again. Wait, the original numerator: 2/x + 1/(3x²). Let's compute that: 2/x is 6x/(3x²), so 6x/(3x²) + 1/(3x²) = (6x + 1)/(3x²). Denominator: 1 + 1/(6x) = (6x + 1)/(6x). Then dividing: (6x + 1)/(3x²) divided by (6x + 1)/(6x) is (6x + 1)/(3x²) * 6x/(6x + 1) = 6x/(3x²) = 2/x. So that's correct. So option A is 2/x, which is the simplified form. But when we plug x = 1, both A and B give 2, x = 2, both give 1. So maybe the problem considers B as a simplified form, but A is more simplified. Wait, but let's check the options again. The options are A. 2/x, B. (12x + 2)/(6x² + x), C. 2x, D. 4/x. So the correct answer is A? But when we multiply numerator and denominator by 6x², we get B, which is also correct. But maybe the problem expects B? Wait, no, let's see the process. When simplifying complex fractions, a common method is to multiply numerator and denominator by the least common denominator of all the fractions in the numerator and denominator. The denominators in the numerator are x and 3x², so LCD is 3x². The denominators in the denominator are 1 (which is x⁰) and 6x, so LCD is 6x. The overall LCD for all denominators (x, 3x², 6x) is 6x². So multiplying numerator and denominator by 6x²:

Numerator: 6x²*(2/x) + 6x²*(1/3x²) = 12x + 2

Denominator: 6x²*1 + 6x²*(1/6x) = 6x² + x

So the expression becomes (12x + 2)/(6x² + x), which is option B. Then, factoring, we get 2(6x + 1)/(x(6x + 1)) = 2/x, which is option A. So both are correct, but maybe the problem expects option B as the intermediate step, or maybe there's a mistake. But since when we plug in values, both A and B give the same result, but the question is to simplify, and 2/x is simpler, so option A? Wait, but let's check the arithmetic again. Wait, 6x divided by 3x²: 6/3 is 2, x/x² is 1/x, so 2/x. Yes, that's correct. So the correct answer is A. 2/x? Wait, but when we did the multiplication by 6x², we got B, which simplifies to A. So maybe the problem has both A and B as correct, but in the options, A is 2/x, which is the simplified form. So the answer is A. 2/x? Wait, no, wait, let's check with x = 3. Original expression: numerator: 2/3 + 1/(3*9) = 2/3 + 1/27 = 18/27 + 1/27 = 19/27; denominator: 1 + 1/(18) = 19/18; so expression is (19/27)/(19/18) = 18/27 = 2/3. Option A: 2/3, which matches. Option B: (36 + 2)/(54 + 3) = 38/57 = 2/3, which also matches. So both are correct. But the question is to simplify, so the most simplified form is 2/x, which is option A.

Answer:

A. $\frac{2}{x}$