simplify the fraction
4t to the second power-16 over 8 over t-2 over 6
t+2/12
((4t^2-16)/8) / (t-2)/6,
Invert and multiply:
((4t^2-16)/8) * (6/(t-2)) =
((2t-4)(2t+4)/8) * (6/(t-2)) =
(2(t-2)2(t+2)/8) * (6/(t-2) =
24(t+2) / 8 = 3(t+2).
To simplify the given fraction, we will first factorize the numerator and denominator separately.
Numerator: 4t^2 - 16
We can start by factoring out a common factor, which is 4:
4t^2 - 16 = 4(t^2 - 4)
Next, we can recognize the numerator as the difference of squares, so we can factor further:
4(t^2 - 4) = 4(t - 2)(t + 2)
Denominator: 8/t - 2/6
To simplify this part, in order to combine the fractions, we need a common denominator.
The least common multiple of 8 and 6 is 24. We multiply the first fraction's numerator and denominator by 3, and the second fraction's numerator and denominator by 4:
(8/t) * (3/3) - (2/6) * (4/4) = (24/3t) - (8/24)
Simplifying the denominator further:
(24/3t) - (8/24) = (24/3t) - (1/3)
Now, we can combine the fractions by finding a common denominator, which is 3t:
(24/3t) - (1/3) = (24 - t)/(3t)
Therefore, the simplified fraction is (4(t - 2)(t + 2))/(3t - 3).