Question

√(23-8√7)+√(23+√7)=?

Answer 1

Ответ: 6

1. 23-8√7= 1

2. 23+√7 = 25

3. √1+√25= 6

Answer 2

To simplify the expression √(23-8√7) + √(23+√7), we can start by breaking it down into smaller parts and simplify each part individually.

Let's simplify √(23-8√7) first.

Let's assume √(23-8√7) = a + b√7, where a and b are real numbers.

Squaring both sides: (23-8√7) = (a + b√7)^2 Expanding the right side: 23 - 8√7 = a^2 + 2ab√7 + 7b^2

By comparing the real and imaginary parts, we get two equations:

1. 23 = a^2 + 7b^2 (equation 1)
2. -8√7 = 2ab√7 (equation 2)

From equation 2, we can simplify it to: -4 = ab

By substituting the value of b from equation 2 into equation 1, we get: 23 = a^2 + 7(-4/a)^2 23 = a^2 + 7(16/a^2) 23 = a^2 + 112/a^2

Now, we have a quadratic equation in terms of a. Multiplying through by a^2: 23a^2 = a^4 + 112

Rearranging the terms: a^4 - 23a^2 + 112 = 0

This is a quadratic equation in terms of a^2. Let's solve this equation using factoring: (a^2 - 7)(a^2 - 16) = 0

Setting each factor equal to zero: a^2 - 7 = 0 --> a^2 = 7 --> a = √7 (taking the positive square root) a^2 - 16 = 0 --> a^2 = 16 --> a = 4 (taking the positive square root)

Since we assumed √(23-8√7) = a + b√7, we have two possibilities for a: a = √7 or a = 4.

If a = √7, we can substitute it back into equation 2: -4 = (√7)b b = -4/√7 = -4√7/7

Therefore, one possible solution is: √(23-8√7) = √7 - (4√7/7) = (3√7)/7

Now, let's simplify √(23+√7).

√(23+√7) cannot be further simplified since there are no common factors.

Finally, we can calculate the sum: √(23-8√7) + √(23+√7) = (3√7)/7 + √(23+√7)