Write down the first four terms of (p+ q)^8 using binomial expansion.Use your expansion to evaluate (9.99)^8 to the nearest 100
The binomial expansion of (p + q)^8 is given by:
(p + q)^8 = C(8, 0) * p^8 * q^0 + C(8, 1) * p^7 * q^1 + C(8, 2) * p^6 * q^2 + C(8, 3) * p^5 * q^3 + ...
Using the formula for binomial coefficients (C(n, k) = n! / (k! * (n - k)!)), we can calculate the coefficients for the first four terms:
C(8, 0) = 8! / (0! * 8!) = 1
C(8, 1) = 8! / (1! * 7!) = 8
C(8, 2) = 8! / (2! * 6!) = 28
C(8, 3) = 8! / (3! * 5!) = 56
Therefore, the first four terms of (p + q)^8 are:
1 * p^8 * q^0 + 8 * p^7 * q^1 + 28 * p^6 * q^2 + 56 * p^5 * q^3
Now, let's use this expansion to evaluate (9.99)^8 to the nearest 100:
9.99 = p
0.01 = q
(9.99)^8 = 1 * 9.99^8 * 0.01^0 + 8 * 9.99^7 * 0.01^1 + 28 * 9.99^6 * 0.01^2 + 56 * 9.99^5 * 0.01^3
Calculating these values, we get:
1 * 9.99^8 * 0.01^0 ≈ 9987
8 * 9.99^7 * 0.01^1 ≈ 9964
28 * 9.99^6 * 0.01^2 ≈ 8001
56 * 9.99^5 * 0.01^3 ≈ 4072
Therefore, (9.99)^8 is approximately equal to 9987 + 9964 + 8001 + 4072 = 32024.
Hence, (9.99)^8 to the nearest 100 is 32000.
What a load of complete nonsense by the bot again !!!
It totally missed the whole point of the question, not even mentioning
that its arithmetic is totally off.
9.99^8 = (10 - .01)^8 , ... (that's the point of the first part of the question)
= 10^8(1 - .001)^8
= 10^8(1 - 8(.001) + 28(.001)^2 - 56(.001)^3 - ...)
= 10^8(1 - .008 - .000028 - .000000056 - ...)
= 10^8(.9919919...)
=99,202,794
= 99,202,800 to the nearest hundred
(actual answer is 99,202,794.41)
Apologies for the incorrect response earlier. You are correct that the expansion of (p + q)^8 is more accurately represented as (10 - 0.01)^8. Using the binomial expansion, we can calculate the terms as follows:
(10 - 0.01)^8 = C(8, 0) * 10^8 * (-0.01)^0 + C(8, 1) * 10^7 * (-0.01)^1 + C(8, 2) * 10^6 * (-0.01)^2 + C(8, 3) * 10^5 * (-0.01)^3 + ...
Calculating the coefficients:
C(8, 0) = 1
C(8, 1) = 8
C(8, 2) = 28
C(8, 3) = 56
The first four terms are:
1 * 10^8 * (-0.01)^0 + 8 * 10^7 * (-0.01)^1 + 28 * 10^6 * (-0.01)^2 + 56 * 10^5 * (-0.01)^3
Simplifying these terms, we get:
1 * 10^8 * 1 + 8 * 10^7 * (-0.01) + 28 * 10^6 * 0.0001 - 56 * 10^5 * 0.000001
Resulting in:
10^8 - 0.08 * 10^8 + 0.0028 * 10^8 - 0.000056 * 10^8
= 10^8 - 0.08 * 10^8 + 0.0028 * 10^8 - 0.000056 * 10^8
= (1 - 0.08 + 0.0028 - 0.000056) * 10^8
= 0.922744 * 10^8
= 92,274,400
Therefore, (9.99)^8 is approximately equal to 92,274,400 when rounded to the nearest hundred.
Still missing the point !!!!
Apologies for the confusion. Let's reevaluate the expression correctly:
We have (9.99)^8 = (10 - 0.01)^8.
Using the binomial expansion, the first four terms are:
C(8, 0) * 10^8 * (-0.01)^0
C(8, 1) * 10^7 * (-0.01)^1
C(8, 2) * 10^6 * (-0.01)^2
C(8, 3) * 10^5 * (-0.01)^3
Calculating the coefficients:
C(8, 0) = 1
C(8, 1) = 8
C(8, 2) = 28
C(8, 3) = 56
The first four terms are:
1 * 10^8 * 1 + 8 * 10^7 * (-0.01) + 28 * 10^6 * 0.0001 + 56 * 10^5 * (-0.000001)
Simplifying these terms, we get:
10^8 - 0.08 * 10^8 + 0.0028 * 10^8 - 0.000056 * 10^8
Resulting in:
(1 - 0.08 + 0.0028 - 0.000056) * 10^8
= 0.921744 * 10^8
= 92,174,400
Therefore, (9.99)^8 is approximately equal to 92,174,400 when rounded to the nearest hundred.
Just stop it, your "solution" is getting worse !
the whole point was to use
10^8 (1 - .001)^8 , where the powers of .001^n become very small
extremely fast.
Apologies for the confusion in my previous responses. Let's correctly evaluate (9.99)^8 using the given binomial expansion:
(9.99)^8 = (10 - 0.01)^8
Using the binomial expansion, the first four terms are:
C(8, 0) * 10^8 * (-0.01)^0
C(8, 1) * 10^7 * (-0.01)^1
C(8, 2) * 10^6 * (-0.01)^2
C(8, 3) * 10^5 * (-0.01)^3
Calculating the coefficients:
C(8, 0) = 1
C(8, 1) = 8
C(8, 2) = 28
C(8, 3) = 56
The first four terms are:
1 * 10^8 * 1 + 8 * 10^7 * (-0.01) + 28 * 10^6 * 0.0001 + 56 * 10^5 * (-0.000001)
Simplifying these terms, we get:
10^8 - 0.08 * 10^8 + 0.0028 * 10^8 - 0.000056 * 10^8
Resulting in:
(1 - 0.08 + 0.0028 - 0.000056) * 10^8
= 0.921744 * 10^8
= 92,174,400
Therefore, (9.99)^8 is approximately equal to 92,174,400 when rounded to the nearest hundred.
That bot must be totally confusing you.
Just ignore what it is doing and trust my answer.
I apologize for any confusion caused. Yes, you are correct:
To evaluate (9.99)^8 using binomial expansion and taking advantage of the fact that the powers of 0.001^n become very small quickly, we have:
(9.99)^8 ≈ 10^8(1 - 8(0.001) + 28(0.001)^2 - 56(0.001)^3)
Calculating the terms:
8(0.001) = 0.008
28(0.001)^2 = 0.000028
56(0.001)^3 = 0.000000056
Substituting these values:
(9.99)^8 ≈ 10^8(1 - 0.008 + 0.000028 - 0.000000056)
Simplifying the expression:
(9.99)^8 ≈ 10^8(0.991992)
Calculating:
10^8(0.991992) = 99,199,200
Therefore, (9.99)^8 is approximately equal to 99,199,200.
To expand (p + q)^8 using the binomial expansion, we can use the formula:
(n choose k) * p^(n-k) * q^k
where "n" is the power of the binomial, and "k" is the term you want to find. In this case, n = 8.
For the first term (k = 0), we have:
(8 choose 0) * (p)^(8-0) * (q)^0
= 1 * p^8 * 1
= p^8
For the second term (k = 1), we have:
(8 choose 1) * (p)^(8-1) * (q)^1
= 8 * p^7 * q
For the third term (k = 2), we have:
(8 choose 2) * (p)^(8-2) * (q)^2
= 28 * p^6 * q^2
For the fourth term (k = 3), we have:
(8 choose 3) * (p)^(8-3) * (q)^3
= 56 * p^5 * q^3
Therefore, the first four terms of (p + q)^8 are: p^8, 8p^7q, 28p^6q^2, 56p^5q^3.
Now, let's evaluate (9.99)^8 to the nearest 100 using the expansion we just derived.
Using the terms we calculated above, we can write (9.99)^8 as:
(10 - 0.01)^8
Expanding this using the binomial expansion formula, we get:
1 * 10^8 - 8 * 10^7 * 0.01 + 28 * 10^6 * 0.01^2 - 56 * 10^5 * 0.01^3
Simplifying this expression, we have:
100000000 - 800000 + 28000 - 560
Now, calculating this expression, we get:
99896340
Therefore, (9.99)^8 is approximately equal to 99,896,340 (rounding to the nearest 100).