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avatar+642 

a) Fill in the blanks with positive integers:

\((3+√5)^3\) = __+ __\(√5 \)

 

b) Find the coefficient of \(y^4 \) in the expansion of \((2y-5)^6\)

 Dec 11, 2023
 #1
avatar+222 
0

First problem:

 

(3 + sqrt(5))^3 is equal to 215 + 106 *sqrt(5).

 

We can expand the cube as follows:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

 

In this case, a = 3 and b = sqrt(5):

(3 + sqrt(5))^3 = 3^3 + 3(3^2)(sqrt(5)) + 3(3)(sqrt(5))^2 + (sqrt(5))^3

= 27 + 27sqrt(5) + 15 + 5

= 215 + 106sqrt(5)

 Dec 11, 2023
 #2
avatar+25 
+1

b: (ax+b)^6 has a x^4 coefficient of 15*a^4*b^2, so in this case a = 2, b = -5, so 15*16*25 = 3*8*2*125 = 6000

 Dec 12, 2023

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