Perform the following operation and express in simplest form.
start fraction, x, squared, minus, 64, divided by, x, minus, 8, end fraction, divided by, start fraction, x, squared, plus, 5, x, minus, 24, divided by, 3, x, end fraction
x−8
x
2
−64
÷
3x
x
2
+5x−24
To solve this problem, we need to perform the operations and then simplify the expression.
The first step is to simplify the numerator:
(x^2 - 64) / (x - 8) = [(x + 8)(x - 8)] / (x - 8) = x + 8
Next, we simplify the denominator:
(x^2 + 5x - 24) / (3x) = (x + 8)(x - 3) / 3x
Now, we can divide the fractions:
(x + 8) / [(x + 8)(x - 3) / 3x] = (x + 8) * (3x) / (x + 8)(x - 3)
The (x + 8) terms cancel out, and we are left with:
3x / (x - 3)
Therefore, the simplified expression is 3x / (x - 3).
To simplify the expression, let's break it down step by step.
1. Simplify the numerator of the first fraction (x^2 - 64). This is a difference of squares, which can be factored as (x - 8)(x + 8).
2. Simplify the denominator of the first fraction (x - 8).
3. Simplify the numerator of the second fraction (x^2 + 5x - 24). This can be factored as (x - 3)(x + 8).
4. Simplify the denominator of the second fraction (3x).
After simplifying, the expression becomes:
[(x - 8)(x + 8)] / (x - 8) ÷ [(x - 3)(x + 8)] / (3x)
Next, we can cancel out the common factors:
[1 * (x + 8)] / 1 ÷ [(x - 3) * (1)] / (3x)
Simplify further:
(x + 8) / 1 ÷ (x - 3) / (3x)
To divide by a fraction, we can multiply by its reciprocal:
(x + 8) / 1 * (3x) / (x - 3)
Now, let's simplify by canceling out the common factors:
3(x + 8) / (x - 3)
This is the simplified form of the expression.
To simplify the given expression, let's break it down step by step:
Step 1: Simplify the numerator of the first fraction, x^2 - 64.
This expression is a difference of squares, which can be factored as (x + 8)(x - 8).
Step 2: Simplify the denominator of the first fraction, x - 8.
Step 3: Simplify the numerator of the second fraction, x^2 + 5x - 24.
This expression can be factored as (x + 8)(x - 3).
Step 4: Simplify the denominator of the second fraction, 3x.
Now we can rewrite the expression as:
[(x + 8)(x - 8)] / (x - 8) ÷ [(x + 8)(x - 3)] / (3x)
When dividing fractions, we can simplify by multiplying the first fraction by the reciprocal of the second fraction:
[(x + 8)(x - 8)] / (x - 8) * (3x) / [(x + 8)(x - 3)]
Now, cancel out the common factors between the numerator and denominator:
= (3x) / (x - 3)
Therefore, the simplified form of the expression is (3x) / (x - 3).