(2+root(3))^6+[(2-root(3)]
without using calcultor
let
a=2+sqrt(3) b=2-sqrt(3)
but
a=((2+sqrt(3)^2)
b=[(2-sqrt(3)^2]
a^2=(2+sqrt(3))^2
=(2+sqrt(3)(2+sqrt(3)=7+4sqrt(3)
b^2=(2-sqrt(2)]^2
=(2-sqrt(3))(2-sqrt(3)
=7-4sqrt(3)
but
(a^3+b^3)=(a+b)(a^2-ab+b^2)
a+b=7+4sqrt(3)+7-4sqrt(3)
a+b=14
ab=(7-4sqr(3)(7+4sqrt(3)
ab=1
a^2=(7+4sqrt(3)(7+4sqrt(3)
=97+56sqrt(3)
b^2=(7-4sqrt(3)^2
b^2=(7-4sqrt(3)(7-4sqrt(3)=97-56sqrt(3)
a^3+b^3=(a+b)(a^2-ab+b^2)
=14(97+56sqrt(3)-1+96-56sqrt(3)
=14(97-1+97)
=14(193)=2702 yeah
To simplify the expression (2+√3)^6+[(2-√3)], we can expand both terms separately and then combine them.
First, let's expand (2+√3)^6 using the binomial theorem:
(2+√3)^6 = C(6,0)(2)^6(√3)^0 + C(6,1)(2)^5(√3)^1 + C(6,2)(2)^4(√3)^2 + C(6,3)(2)^3(√3)^3 + C(6,4)(2)^2(√3)^4 + C(6,5)(2)^1(√3)^5 + C(6,6)(2)^0(√3)^6
Simplifying each term:
= 1(2^6) + 6(2^5)(√3) + 15(2^4)(3) + 20(2^3)(3√3) + 15(2^2)(3^2) + 6(2)(3^2√3) + 1(3^3)
= 64 + 192√3 + 720 + 960√3 + 540 + 72√3 + 27
= 64 + 720 + 540 + 27 + (192√3 + 960√3 + 72√3)
= 1351 + (1224√3)
Next, let's simplify (2-√3):
(2-√3)
Finally, we can substitute the simplified values back into the original expression:
(2+√3)^6 + [(2-√3)]
= 1351 + (1224√3) + (2-√3)
= 1353 + 1223√3
Therefore, the simplified expression is 1353 + 1223√3.
To simplify the expression (2+√3)^6+[(2-√3)], we can use the binomial theorem, which states that for any positive integer n:
(a+b)^n = C(n, 0)a^n + C(n, 1)a^(n-1)b + C(n, 2)a^(n-2)b^2 + ... + C(n, n-1)ab^(n-1) + C(n, n)b^n
Where C(n, k) represents the binomial coefficient, given by:
C(n, k) = n! / (k! * (n-k)!)
Let's calculate the expression step by step:
Step 1: Calculating the binomial coefficients.
We need to calculate the binomial coefficients for each term in the expansion. For example, for (2+√3)^6, we have:
C(6, 0) = 6! / (0! * 6!) = 1
C(6, 1) = 6! / (1! * 5!) = 6
C(6, 2) = 6! / (2! * 4!) = 15
C(6, 3) = 6! / (3! * 3!) = 20
C(6, 4) = 6! / (4! * 2!) = 15
C(6, 5) = 6! / (5! * 1!) = 6
C(6, 6) = 6! / (6! * 0!) = 1
Step 2: Expanding the terms.
We expand each term in the binomial expression:
(2+√3)^6 = 1*(2^6) + 6*(2^5)*(√3) + 15*(2^4)*(√3)^2 + 20*(2^3)*(√3)^3 + 15*(2^2)*(√3)^4 + 6*2*(√3)^5 + 1*(√3)^6
(2-√3) = 2 - √3
Step 3: Simplifying the terms.
We simplify each term by calculating the powers of 2 and √3:
= 1*64 + 6*32*√3 + 15*16*3 + 20*8*√3^3 + 15*4*√3^4 + 6*2*√3^5 + 1*√3^6 + (2 - √3)
= 64 + 192√3 + 720 + 160√3 + 60√3^2 + 12√3^3 + √3^4 + 2 - √3
Step 4: Simplifying further.
Now, let's simplify the terms involving √3:
= 786 + (192+160)*√3 + 60*3 + 12√3 + √3^2 + 2 - √3
= 786 + 352√3 + 180 + 12√3 + 3 + 2 - √3
= 971 + 364√3 - √3
= 971 + 363√3
So, the simplified form of the expression (2+√3)^6+[(2-√3)] without using a calculator is 971 + 363√3.