Question

how can ari simplify the following expression?

write the numerator and denominator with a common denominator. then divide the numerator by the denominator. to do this, multiply the numerator by the reciprocal of the denominator.

divide the numerator and the denominator by ( a - 3 ). then divide the numerator by the denominator.

divide the numerator and the denominator by ( a - 3 ). then simplify the numerator and simplify the denominator.

write the numerator and denominator with a common denominator. then divide the numerator by the denominator. to do this, multiply the numerators and multiply the denominators.

Explanation:

Step1: Analyze the expression

The given expression is a complex fraction: $\frac{\frac{a - 4}{a - 3}}{2+\frac{1}{a - 3}}$. First, simplify the denominator of the complex fraction. The denominator is $2+\frac{1}{a - 3}$. To combine these, find a common denominator, which is $a - 3$. So, $2=\frac{2(a - 3)}{a - 3}$, then $2+\frac{1}{a - 3}=\frac{2(a - 3)+1}{a - 3}=\frac{2a-6 + 1}{a - 3}=\frac{2a-5}{a - 3}$.

Step2: Rewrite the complex fraction

Now the complex fraction becomes $\frac{\frac{a - 4}{a - 3}}{\frac{2a-5}{a - 3}}$. When dividing by a fraction, we multiply by its reciprocal. So, this is equivalent to $\frac{a - 4}{a - 3}\times\frac{a - 3}{2a-5}$.

Step3: Simplify the expression

The $(a - 3)$ terms in the numerator and denominator cancel out (assuming $a
eq3$), leaving $\frac{a - 4}{2a-5}$.

Looking at the options, the correct approach should involve simplifying the denominator first (finding a common denominator), then using the reciprocal of the denominator to multiply (since dividing by a fraction is multiplying by its reciprocal). The fourth option says: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator." Wait, actually, let's re - check the options. Wait, maybe I misread. Wait, the first option: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator." Wait, no, let's re - express the steps. The process of simplifying a complex fraction: first, simplify the numerator and denominator (if they are fractions) by getting a common denominator, then divide the numerator fraction by the denominator fraction, which is multiplying the numerator fraction by the reciprocal of the denominator fraction.

Wait, the fourth option's description: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerators and multiply the denominators." No, that's not correct. Wait, maybe the first option? Wait, no, let's re - look at the options.

Wait, the correct method for simplifying $\frac{\frac{a - 4}{a - 3}}{2+\frac{1}{a - 3}}$:

1. Simplify the denominator: $2+\frac{1}{a - 3}=\frac{2(a - 3)+1}{a - 3}=\frac{2a-6 + 1}{a - 3}=\frac{2a-5}{a - 3}$ (this is writing the denominator with a common denominator).

2. Now the expression is $\frac{\frac{a - 4}{a - 3}}{\frac{2a-5}{a - 3}}$, which is equal to $\frac{a - 4}{a - 3}\div\frac{2a-5}{a - 3}=\frac{a - 4}{a - 3}\times\frac{a - 3}{2a-5}$ (multiplying by the reciprocal of the denominator).

So the correct option should be the one that describes: first, write the numerator and denominator (of the complex fraction) with common denominators (for the denominator of the complex fraction, we find a common denominator to simplify it), then divide the numerator by the denominator (by multiplying by the reciprocal). Looking at the options, the fourth option: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator." Wait, no, the fourth option's text: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerators and multiply the denominators." No, that's incorrect. Wait, maybe the first option: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator." Wait, the first option's text: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator." Yes, that's correct. Because when you have a complex fraction $\frac{\frac{A}{B}}{\frac{C}{D}}$, you can think of it as $\frac{A}{B}\div\frac{C}{D}=\frac{A}{B}\times\frac{D}{C}$, which is multiplying the numerator by the reciprocal of the denominator. And to get $\frac{C}{D}$ (the denominator of the complex fraction) in a simplified form, you may need to find a common denominator if it's a sum or difference of fractions.

Wait, maybe I made a mistake earlier. Let's re - examine the options:

Option 1: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator."

Option 2: "Divide the numerator and the denominator by $a - 3$. Then divide the numerator by the denominator."

Option 3: "Divide the numerator and the denominator by $a - 3$. Then simplify the numerator and simplify the denominator."

Option 4: "Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerators and multiply the denominators."

The correct approach is:

1. Simplify the denominator of the complex fraction (which is a sum of a whole number and a fraction) by finding a common denominator.

2. Then, since the complex fraction is a fraction divided by a fraction, we divide the numerator fraction by the denominator fraction, which is equivalent to multiplying the numerator fraction by the reciprocal of the denominator fraction.

So Option 1 is correct. Because when you have $\frac{\frac{a - 4}{a - 3}}{2+\frac{1}{a - 3}}$, first, the denominator $2+\frac{1}{a - 3}$ needs a common denominator (which is $a - 3$) to be written as a single fraction. Then, to divide $\frac{a - 4}{a - 3}$ by $\frac{2(a - 3)+1}{a - 3}$, you multiply $\frac{a - 4}{a - 3}$ by the reciprocal of $\frac{2(a - 3)+1}{a - 3}$, which is $\frac{a - 3}{2(a - 3)+1}$.

Answer:

A. Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator.