Using the binomial theroem to estimate 6.07^2
ive got 6^2(1+0.01)^2
but im not sure if its right and i don't know where to go for here
You will have to know the special case of the general binomial expansion, namely
(1 + x)^n = 1 + nx + n(n-1)/2!x^2 + n(n-1)(n-2)/3!x^3 + ..
if │x│ < 1 , then the terms become rapidly smaller and less significant as you continue down the series
so 6.07^2
= 6^2(1 + .011667)^2
= 36[1 + 2(.011667 + .011667^2) + 0 + 0 + ...]
(notice if n=2, then one of the factors in n(n-1)(n-2).. becomes zero and all subsequent terms are zero)
so now evaluate 36(1 + 2(.011667)
to get 36.84
PS:
This is a very silly question. The calculations needed for your estimation are more difficult than those needed to just multiply 6.07 x 6.07
The binomial theorem was an important tool in finding roots and huge powers when we had no calculators, but to use it in such a question as the one you posted makes very little sense
To estimate 6.07^2 using the binomial theorem, the first step is to rewrite it as (6 + 0.07)^2.
The binomial theorem allows us to expand the expression (a + b)^n for any exponent n. In this case, a = 6 and b = 0.07. The exponent is 2.
Using the formula, the expansion of (a + b)^n is given by:
(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n
where C(n, k) represents the binomial coefficient, given by:
C(n, k) = n! / (k! * (n-k)!)
Let's apply this formula to our expression (6 + 0.07)^2:
(6 + 0.07)^2 = C(2, 0) * 6^2 * 0.07^0 + C(2, 1) * 6^1 * 0.07^1 + C(2, 2) * 6^0 * 0.07^2
Now, let's calculate each term:
C(2, 0) = 2! / (0! * (2-0)!) = 1
C(2, 1) = 2! / (1! * (2-1)!) = 2
C(2, 2) = 2! / (2! * (2-2)!) = 1
Plugging these values into the expansion:
(6 + 0.07)^2 = 1 * 6^2 * 0.07^0 + 2 * 6^1 * 0.07^1 + 1 * 6^0 * 0.07^2
Simplifying the calculations:
(6 + 0.07)^2 = 1 * 36 * 1 + 2 * 6 * 0.07 + 1 * 1 * 0.0049
= 36 + 0.84 + 0.0049
= 36.8449
So, using the binomial theorem, the estimated value of 6.07^2 is approximately 36.8449.