Advertisements
Advertisements
Question
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Solution
`(a + b)^6 = ^6C_0 a^6 + ^6C_1 a^5 b + ^6C_2 a^4 b^2 + ^6C_3 a^3 b^3 + ^6C_4 a^2 b^4 + ^6C_5a^1b^5 + ^6C_6 b^6`
= `a^6 + 6a^5b + 15a^4 b^2 + 20a^3 b^3 + 15a^2 b^4 + 6ab^5 + b^6`
`(a - b)^6 = ^6C_0 a^6 - ^6C_1 a^5 b + ^6C_2 a^4 b^2 - ^6C_3 a^3 b^3 + ^6C_4 a^2 b^4 - ^6C_5a^1b^5 + ^6C_6 b^6`
= `a^6 - 6a^5b + 15a^4 b^2 - 20a^3 b^3 + 15a^2 b^4 - 6ab^5 + b^6`
∴ `(a + b)^6 - (a -b)^6 = 2(6a^5b + 20a^3 b^3 + 6ab^5)`
Putting a = `sqrt3` and b = `sqrt2`, we obtain
`(sqrt3 + sqrt2)^6 - (sqrt3 + sqrt2)^6` = `2[6(sqrt3)^5 (sqrt2) + 20 (sqrt3)^3 (sqrt2)^3 + 6 (sqrt3)(sqrt2)^5]`
= `2[54sqrt6 + 120 sqrt6 + 24 sqrt6]`
= `2 xx 198 sqrt6`
= `396 sqrt6`
APPEARS IN
RELATED QUESTIONS
Using Binomial Theorem, evaluate of the following:
(102)5
Using binomial theorem, evaluate the following:
(99)5
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer.
[Hint: write an = (a – b + b)n and expand]
Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Expand the following (1 – x + x2)4
Find the 4th term from the end in the expansion of `(x^3/2 - 2/x^2)^9`
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x10?
Which of the following is larger? 9950 + 10050 or 10150
Find the coefficient of x50 after simplifying and collecting the like terms in the expansion of (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + ... + x1000 .
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is ______.
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.
Find the coefficient of x15 in the expansion of (x – x2)10.
If the coefficient of second, third and fourth terms in the expansion of (1 + x)2n are in A.P. Show that 2n2 – 9n + 7 = 0.
Given the integers r > 1, n > 2, and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then ______.
The number of terms in the expansion of (x + y + z)n ______.
The coefficient of a–6b4 in the expansion of `(1/a - (2b)/3)^10` is ______.
Number of terms in the expansion of (a + b)n where n ∈ N is one less than the power n.
The positive integer just greater than (1 + 0.0001)10000 is ______.
