Advertisements
Advertisements
Question
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Advertisements
Solution
Using Binomial Theorem, the expressions, (a + b)4 and (a – b)4, can be expanded as
`(a + b)^4 = ^4C_0 a^4 + ^4C_1 a^3 b + ^4C_2 a^2b^2 + ^4C_3 ab^3 + ^4C_4 b^4`
(a - b)4 = 4C0 a4 - 4C1 a3b + 4C2 a2b2 - 4C3 ab3 + 4C4b4
∴ `(a + b)^4 - (a - b)^4 = ^4C_0 a^4 + ^4C_1 a^3 b + ^4C_2 a^2b^2 + ^4C_3 ab^3 + ^4C_4 b^4`
[4C0 a4 - 4C1 a3b + 4C2 a2b2 - 4C3 ab3 + 4C4 b4]
2 (4C1a3b + 4C3ab3) = 2(4a3b + 4ab3)
= 8ab (a2 + b2)
In this, by substituting `a = sqrt 3 , b = sqrt 2`
`(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
= `8sqrt3. sqrt2 [(sqrt3)^2 + (sqrt2)^2]`
= `8sqrt6 (3 + 2) = 40sqrt6`
APPEARS IN
RELATED QUESTIONS
Expand the expression: (1– 2x)5
Expand the expression: (2x – 3)6
Expand the expression: `(x + 1/x)^6`
Using Binomial Theorem, evaluate the following:
(96)3
Using Binomial Theorem, evaluate of the following:
(102)5
Using binomial theorem, evaluate f the following:
(101)4
Using binomial theorem, evaluate the following:
(99)5
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
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 an approximation of (0.99)5 using the first three terms of its expansion.
Expand using Binomial Theorem `(1+ x/2 - 2/x)^4, x != 0`
Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Find the rth term in the expansion of `(x + 1/x)^(2r)`
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 (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
Find the coefficient of x15 in the expansion of (x – x2)10.
Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.
Find the coefficient of x4 in the expansion of (1 + x + x2 + x3)11.
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that O2 – E2 = (x2 – a2)n
The total number of terms in the expansion of (x + a)100 + (x – a)100 after simplification is ______.
The number of terms in the expansion of (x + y + z)n ______.
Let the coefficients of x–1 and x–3 in the expansion of `(2x^(1/5) - 1/x^(1/5))^15`, x > 0, be m and n respectively. If r is a positive integer such that mn2 = 15Cr, 2r, then the value of r is equal to ______.
Let `(5 + 2sqrt(6))^n` = p + f where n∈N and p∈N and 0 < f < 1 then the value of f2 – f + pf – p is ______.
