(4x+6)/(16x^2-36)/(4x^2+22x-42)
To simplify the expression (4x+6)/(16x^2-36)/(4x^2+22x-42), we can follow these steps:
Step 1: Simplify the individual expressions inside the numerator and denominator of the main fraction.
The numerator, (4x + 6), cannot be simplified further.
For the denominator:
- Factorize both expressions:
- 16x^2 - 36 can be factored as 4(4x^2 - 9), and we recognize it as the difference of squares.
- 4x^2 + 22x - 42 can be factored as (4x - 6)(x + 7).
Step 2: Rewrite the expression with the simplified numerator and denominator.
The expression becomes (4x + 6) / [4(4x^2 - 9) / (4x - 6)(x + 7)].
Step 3: Simplify further by canceling out common factors or terms.
In this case, observe that there is a common factor of 2 between the numerator (4x + 6) and the denominator (4x - 6). By canceling out this common factor, the expression simplifies to:
(2x + 3) / [2(4x^2 - 9) / (x + 7)].
And that's the final simplified form of the given expression:
(2x + 3) / [2(4x^2 - 9) / (x + 7)].