Answer :
Let's work through the expression step-by-step to simplify it.
We have two fractions being multiplied:
[tex]\[ \frac{8x^2 - 24x}{16x^3 - 48x^2} \cdot \frac{40x^3 + 56x^2}{5x^2 - 43x - 70} \][/tex]
### Step 1: Simplify each fraction
First Fraction: [tex]\(\frac{8x^2 - 24x}{16x^3 - 48x^2}\)[/tex]
1. Factor out the greatest common factor (GCF) in the numerator:
- The GCF of [tex]\(8x^2\)[/tex] and [tex]\(24x\)[/tex] is [tex]\(8x\)[/tex].
- So, [tex]\(8x^2 - 24x = 8x(x - 3)\)[/tex].
2. Factor out the GCF in the denominator:
- The GCF of [tex]\(16x^3\)[/tex] and [tex]\(48x^2\)[/tex] is [tex]\(16x^2\)[/tex].
- So, [tex]\(16x^3 - 48x^2 = 16x^2(x - 3)\)[/tex].
3. Simplify the fraction by cancelling out common factors:
- The [tex]\((x - 3)\)[/tex] and [tex]\(8x\)[/tex] in the numerator can be cancelled with [tex]\(16x^2(x - 3)\)[/tex] in the denominator, leaving us with [tex]\(\frac{1}{2x}\)[/tex].
Second Fraction: [tex]\(\frac{40x^3 + 56x^2}{5x^2 - 43x - 70}\)[/tex]
1. Factor the numerator:
- The GCF of [tex]\(40x^3\)[/tex] and [tex]\(56x^2\)[/tex] is [tex]\(8x^2\)[/tex].
- So, [tex]\(40x^3 + 56x^2 = 8x^2(5x + 7)\)[/tex].
2. Factor the denominator:
- Factors of [tex]\(-70\)[/tex] that add up to [tex]\(-43\)[/tex] are [tex]\(-50\)[/tex] and [tex]\(7\)[/tex].
- So, [tex]\(5x^2 - 43x - 70 = (5x + 7)(x - 10)\)[/tex].
### Step 2: Multiply the simplified fractions
Now, multiply the simplified fractions:
[tex]\[ \frac{1}{2x} \cdot \frac{8x^2(5x + 7)}{(5x + 7)(x - 10)} \][/tex]
1. Cancel out common factors:
- [tex]\((5x + 7)\)[/tex] can be canceled from the numerator and the denominator.
This leaves us with:
[tex]\[ \frac{4x}{x - 10} \][/tex]
So, the simplified result is:
[tex]\[ \boxed{\frac{4x}{x - 10}} \][/tex]
We have two fractions being multiplied:
[tex]\[ \frac{8x^2 - 24x}{16x^3 - 48x^2} \cdot \frac{40x^3 + 56x^2}{5x^2 - 43x - 70} \][/tex]
### Step 1: Simplify each fraction
First Fraction: [tex]\(\frac{8x^2 - 24x}{16x^3 - 48x^2}\)[/tex]
1. Factor out the greatest common factor (GCF) in the numerator:
- The GCF of [tex]\(8x^2\)[/tex] and [tex]\(24x\)[/tex] is [tex]\(8x\)[/tex].
- So, [tex]\(8x^2 - 24x = 8x(x - 3)\)[/tex].
2. Factor out the GCF in the denominator:
- The GCF of [tex]\(16x^3\)[/tex] and [tex]\(48x^2\)[/tex] is [tex]\(16x^2\)[/tex].
- So, [tex]\(16x^3 - 48x^2 = 16x^2(x - 3)\)[/tex].
3. Simplify the fraction by cancelling out common factors:
- The [tex]\((x - 3)\)[/tex] and [tex]\(8x\)[/tex] in the numerator can be cancelled with [tex]\(16x^2(x - 3)\)[/tex] in the denominator, leaving us with [tex]\(\frac{1}{2x}\)[/tex].
Second Fraction: [tex]\(\frac{40x^3 + 56x^2}{5x^2 - 43x - 70}\)[/tex]
1. Factor the numerator:
- The GCF of [tex]\(40x^3\)[/tex] and [tex]\(56x^2\)[/tex] is [tex]\(8x^2\)[/tex].
- So, [tex]\(40x^3 + 56x^2 = 8x^2(5x + 7)\)[/tex].
2. Factor the denominator:
- Factors of [tex]\(-70\)[/tex] that add up to [tex]\(-43\)[/tex] are [tex]\(-50\)[/tex] and [tex]\(7\)[/tex].
- So, [tex]\(5x^2 - 43x - 70 = (5x + 7)(x - 10)\)[/tex].
### Step 2: Multiply the simplified fractions
Now, multiply the simplified fractions:
[tex]\[ \frac{1}{2x} \cdot \frac{8x^2(5x + 7)}{(5x + 7)(x - 10)} \][/tex]
1. Cancel out common factors:
- [tex]\((5x + 7)\)[/tex] can be canceled from the numerator and the denominator.
This leaves us with:
[tex]\[ \frac{4x}{x - 10} \][/tex]
So, the simplified result is:
[tex]\[ \boxed{\frac{4x}{x - 10}} \][/tex]
