In the expansion of (ax +by)^8, the first three terms are 256x^8, 6144x^7y and 64512x^6y^2. Find the values of a and b
a=256^1/8=2
2^7*b^1=6144, so b=6144/128=48
To find the values of a and b, we can use the binomial expansion formula.
The binomial expansion formula is given by:
(a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCr * a^(n-r) * b^r + ... + nCn * a^0 * b^n
Given that the first three terms of the expansion are 256x^8, 6144x^7y, and 64512x^6y^2, we can write the expansion as:
(a + b)^8 = 256x^8 + 6144x^7y + 64512x^6y^2 + ...
Comparing the coefficients of like terms, we can create a system of equations:
1. nCr * a^(n-r) * b^r = 256x^8
Plug in n = 8 and r = 8 to get: a^(8-8) * b^8 = 256x^8
Simplify to: b^8 = 256x^8
2. nCr * a^(n-r) * b^r = 6144x^7y
Plug in n = 8 and r = 7 to get: a^(8-7) * b^7 = 6144x^7y
Simplify to: a * b^7 = 6144x^7y
3. nCr * a^(n-r) * b^r = 64512x^6y^2
Plug in n = 8 and r = 6 to get: a^(8-6) * b^6 = 64512x^6y^2
Simplify to: a^2 * b^6 = 64512x^6y^2
Now we have a system of equations:
b^8 = 256x^8 ---(1)
a * b^7 = 6144x^7y ---(2)
a^2 * b^6 = 64512x^6y^2 ---(3)
From equation (1), we can take the 8th root of both sides to get:
b = (256x^8)^(1/8)
b = 2x
Substitute this into equation (2):
a * (2x)^7 = 6144x^7y
a * 128x^7 = 6144x^7y
a = (6144x^7y) / (128x^7)
a = 48y
Substitute the values of a and b into equation (3):
(48y)^2 * (2x)^6 = 64512x^6y^2
2304x^6y^2 = 64512x^6y^2
Simplifying, we get:
2304 = 64512
This is not a valid equation, so the given terms do not provide consistent values for a and b.
To find the values of a and b, we can use the properties of binomial expansion.
The formula for the expansion of (ax + by)^n, where n is a positive integer, is given by:
(ax + by)^n = C(n, 0)(ax)^n + C(n, 1)(ax)^(n-1)(by) + C(n, 2)(ax)^(n-2)(by)^2 + ... + C(n, r)(ax)^(n-r)(by)^r + ... + C(n, n)(by)^n
where C(n, r) represents the binomial coefficient. The binomial coefficient C(n, r) is given by the formula:
C(n, r) = n! / (r!(n-r)!)
Using the given information, we can compare the coefficients of the first three terms in the expansion of (ax + by)^8 to find the values of a and b.
The first term is 256x^8. This means that the expansion of (ax + by)^8 has a term with a coefficient of 256 and a power of x^8. Since this term only involves x variables and not y variables, we know that the coefficient of y in this term is 0. Using the formula for the first term of the binomial expansion, we have:
C(8, 0)(ax)^8 = 256x^8
Simplifying the equation, we can cancel out the factors and solve for a:
1*a^8 = 256
Taking the 8th root of both sides, we find:
a = 2
The second term is 6144x^7y. This means that the expansion of (ax + by)^8 has a term with a coefficient of 6144, a power of x^7, and a power of y^1. Using the formula for the second term of the binomial expansion, we have:
C(8, 1)(ax)^7(by) = 6144x^7y
Simplifying the equation, we can cancel out the factors and solve for a and b:
8*a^7*b = 6144
Since we know that a = 2, we can substitute it into the equation:
8*(2^7)*b = 6144
Simplifying further, we have:
8*128*b = 6144
1024b = 6144
Dividing both sides by 1024, we find:
b = 6
Therefore, the values of a and b are a = 2 and b = 6, respectively.