Question

Oblicz:

1) (√5 + √3)²

2) (4√3 + 5√2)²

3) (3√2 - √3)(3√2+√3)

4) (2√3 - √5)(2√3 +√5)

5)

6)

7) (√2-1)² - (√2 +1)²

8) (1+√3)² - (1 - √3)²

Answer 1

Stosujemy wzory skróconego mnożenia:

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

(a-b)^2 = a^2 - 2ab - b^2 

(a+b)(a-b) = a^2 - b^2

 

1.

(V5+V3)^2 = V5^2 +2V5*V3 + V3^2 = 5+2V5+3 = 8+2V5

2.

(4V3+5V2)^2 = (4V3)^2 + 2 *4V3*V2 +(5V2)^2 = 48+40V6+50 = 98+40V6

3.

(3V2-V3)(3V2+V3) = (3V2)^2 -V3^2 = 18-3 =15

4.

(2V3-V5)(2V3+V5) = 12-5 = 7

5.

[V(2-V3) - V(2+V3) = 2-V3-2V[(2-V3)(2+V3)] + 2 + V3 = 4-2V(4-3) = 4-2*1 = 2

6.

[V(V2-1) = V(V2+1)]^2 = V2-1+2V[(V2-1)(V2+1)] = V2 +1 = 2V2 + 2V(2-1) =

= 2V2 + 2V1 = 2V2 + 2*1 = 2V2 + 2

7.

(V2-1)^2 -(V2+1)^2 = 2-2V2+1 -(2+2V2+1) = 2-2V2+1-2-2V2-1 = -4V2

8.

(1+V3)^2 -(1-3)^2 = 1+2V3+3-(1-2V3+3) = 1+2V3+3-1+2V3-3 = 4V3