How can Ari simplify the following expression? mc025-1.jpg 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. 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. 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
The Correct Answer and Explanation is:
To accurately help you simplify the expression in “mc025-1.jpg”, I would need to see the image. Since the image hasn’t been uploaded or described in your message, I’ll assume a common type of expression that fits the context you’re asking about — one that involves rational expressions and division, possibly like this: a2−9a2−a−6a+3a−3\frac{\frac{a^2 – 9}{a^2 – a – 6}}{\frac{a + 3}{a – 3}}
Let’s simplify this step-by-step:
Step 1: Recognize the Structure
This is a complex fraction — a fraction divided by another fraction: a2−9a2−a−6a+3a−3\frac{\frac{a^2 – 9}{a^2 – a – 6}}{\frac{a + 3}{a – 3}}
To simplify, we multiply the numerator by the reciprocal of the denominator: =a2−9a2−a−6⋅a−3a+3= \frac{a^2 – 9}{a^2 – a – 6} \cdot \frac{a – 3}{a + 3}
Step 2: Factor All Expressions
Factor each polynomial:
- a2−9=(a−3)(a+3)a^2 – 9 = (a – 3)(a + 3)
- a2−a−6=(a−3)(a+2)a^2 – a – 6 = (a – 3)(a + 2)
So now the expression becomes: (a−3)(a+3)(a−3)(a+2)⋅a−3a+3\frac{(a – 3)(a + 3)}{(a – 3)(a + 2)} \cdot \frac{a – 3}{a + 3}
Step 3: Cancel Common Factors
Now cancel out common factors:
- (a−3)(a – 3) appears in numerator and denominator
- (a+3)(a + 3) appears in numerator and denominator
After cancellation: 1a+2\frac{1}{a + 2}
Final Answer:
1a+2\boxed{\frac{1}{a + 2}}
Explanation:
To simplify a complex rational expression, we follow a series of methodical steps. The given expression involves one fraction divided by another. The key principle to apply is: when dividing by a fraction, multiply by its reciprocal.
Starting with the expression: a2−9a2−a−6a+3a−3\frac{\frac{a^2 – 9}{a^2 – a – 6}}{\frac{a + 3}{a – 3}}
we first rewrite the division as multiplication by the reciprocal: a2−9a2−a−6⋅a−3a+3\frac{a^2 – 9}{a^2 – a – 6} \cdot \frac{a – 3}{a + 3}
Before multiplying, it’s helpful to factor all polynomials, which simplifies canceling later. The numerator a2−9a^2 – 9 is a difference of squares, factoring to (a−3)(a+3)(a – 3)(a + 3). The denominator a2−a−6a^2 – a – 6 factors as (a−3)(a+2)(a – 3)(a + 2).
Now the expression is: (a−3)(a+3)(a−3)(a+2)⋅a−3a+3\frac{(a – 3)(a + 3)}{(a – 3)(a + 2)} \cdot \frac{a – 3}{a + 3}
We can now cancel terms common to the numerator and denominator across the entire expression:
- One a−3a – 3 cancels from numerator and denominator.
- One a+3a + 3 cancels similarly.
We’re left with: 1a+2\frac{1}{a + 2}
This is the simplest form of the original complex fraction. Using reciprocal multiplication and factoring made the process clean and efficient. This method is essential in algebra when working with rational expressions.
