How is -16/(4x+6)^5 and -1/(2(2x+3)^5) equivalent?
I think -1/2(2x+3)^5 is the more simplified version of the two, but I just can't figure out how -16/(4x+6)^5 simplifies to -1/(2(2x+3)^5). please explain
When you factor the denominator you get
(4x+6)^5
= 2^5 (2x + 3)^5
= 32(2x+3)^5
so -16/(4x+6)^5
= -16/(32(2x+3)^5)
= -1/(2(2x+3)^5)
further explanation:
(4x+6)^5
= (4x+6)(4x+6)(4x+6)(4x+6)(4x+6)
= (2)(2x+3)(2)(2x+3)(2)(2x+3)(2)(2x+3)(2)(2x+3)
= 32(2x+3)^5
thanks
To understand how these expressions are equivalent, let's start by simplifying the first expression, -16/(4x+6)^5.
1. Start by factoring out a common factor from both the numerator (top) and denominator (bottom). In this case, we can factor out -1 from the numerator:
-16/(4x+6)^5 = (-1 * 16)/(4x+6)^5
2. Simplify the numerator:
(-1 * 16) = -16
3. Simplify the denominator:
The denominator, (4x+6)^5, does not have any common factors that can be simplified further.
Therefore, the simplified form of -16/(4x+6)^5 is -16/(4x+6)^5.
Now, let's compare this result with -1/(2(2x+3)^5).
1. Start by simplifying the denominator, 2(2x+3)^5:
We can simplify the expression 2(2x+3)^5 by multiplying 2 by (2x+3)^5:
2(2x+3)^5 = 2 * (2x+3)^5 = (2 * 2) * (2x+3)^5 = 4(2x+3)^5
Therefore, -1/(2(2x+3)^5) can be rewritten as -1/(4(2x+3)^5).
Comparing this result with the simplified form of -16/(4x+6)^5, we can see that they are indeed equivalent:
-1/(4(2x+3)^5) = -16/(4x+6)^5.
Both expressions represent the same mathematical relationship, just written in different forms.