(4x+6)/(16x^2-36)/(4x^2+22x-42)
(4x+6)/(4x-6)(4x+6)/(2(x+7)(2x-3)
=
1/(4x-6)/(2(x+7)(2x-3)
=
1/(4(2x-3x)(2x^2+11x-21)
unless you meant
(4x+6) / ((16x^2-36)/(4x^2+22x-42))
Then you have
(4x+6) * (x+7)(4x-6) / (4x-6)(4x+6)
= (x+7)
To simplify the expression (4x+6)/(16x^2-36)/(4x^2+22x-42), we need to follow the order of operations, which is PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
1. First, let's simplify the numerator and denominator of the fraction by factoring and canceling common factors, if possible.
Numerator: 4x + 6
The GCF (greatest common factor) of 4 and 6 is 2, so we can factor out 2:
2(2x + 3)
Denominator: 16x^2 - 36
This is a difference of squares, so we can factor it as:
(4x + 6)(4x - 6)
Divisor: 4x^2 + 22x - 42
This can be factored as:
(2x - 2)(2x + 12)
2. Now, we can rewrite the expression with the factored numerator and denominator:
[2(2x + 3)] / [(4x + 6)(4x - 6)] / [(2x - 2)(2x + 12)]
3. Next, since division is the same as multiplication by the reciprocal, we can rewrite the expression as:
[2(2x + 3)] * [(2x - 2)(2x + 12)] / [(4x + 6)(4x - 6)]
4. Then, we can cancel any common factors between the numerator and denominator. In this case, we have (2x + 3) in both the numerator and the denominator, so we can cancel it out:
[2 * 1 * (2x - 2)(2x + 12)] / [(4x + 6)(4x - 6) * 1]
5. Finally, we multiply the remaining terms together to simplify the expression:
[2(2x - 2)(2x + 12)] / [(4x + 6)(4x - 6)]
And that's the simplified form of the expression (4x+6)/(16x^2-36)/(4x^2+22x-42).