Question

3^√20+14 √2 + 3^√20-14 √2

Answer 1

14√2 и -14√2 как бы сокращаются,и остается 6 √20 =12√5

Answer 2

3^√20+14 √2 + 3^√20-14 √2 = 3^√20 + 3^√20 = 6^√20 = 6√(2*2*5)=12√5

Answer 3

To simplify this expression, we can recognize that √20 can be simplified to 2√5. Then, we can use the fact that a^√b + a^-√b = 2, where a > 0 and b > 0.

Let a = 3 and b = 20 - 14√2. Then,

3^√20+14√2 + 3^√20-14√2 = 3^(2√5) + 3^(2√5-28√2) = 3^(2√5) + 3^(-2√5+28√2) [using the fact that a^√b + a^-√b = 2] = (3^(√5))^2 + (3^(-√5))^2(28√2)^2 = (3^(√5))^2 + (3^(-√5))^2(784) = (3^(√5))^2 + (3^(-√5))^2(16^2)

Letting x = 3^√5 and y = 3^(-√5), we have:

(3^(√5))^2 + (3^(-√5))^2(16^2) = x^2 + y^2(16^2) = x^2 + y^2(256) = x^2 + (1/x)^2(256) [using the fact that x*y = 1] = x^2 + (256/x)^2 = (x + 256/x)^2 - 2(256)

Therefore, the simplified expression is:

3^√20+14√2 + 3^√20-14√2 = (3^(√5))^2 + (3^(-√5))^2(16^2) = (x + 256/x)^2 - 2(256) = (3^√5 + 256/3^√5)^2 - 512.