Question

doprowadź do postaci: a+b√c, gdzie: a,b ∈ C, c ∈N:

a) (2- √3)(3+√3)

b) (2√5-1)(3+√5)

c) -√2(3-√2)

d) (7-√3)(7+√3)

e) (2√6-3)(√6+4)

f) (√7-2√3)(2√7+√3)

g) (√2+√3)(3√2-√3)

h(3-4√10)¹⁰

Answer 1

doprowadź do postaci: a+b√c, gdzie: a,b ∈ C, c ∈N:

a) (2- √3)(3+√3)=

= 2*3 +2*√3 -3*√3 -√3*√3 =

= 6 +2√3 -3√3 -3 =

= 3 -√3

 

b) (2√5-1)(3+√5)=

= 3*2√5 + 2√5*√5 - 1*3 -1*√5 =

= 6√5 + 2*5  -3 -√5 =

= 5√5 -7 =

= -7 +5√5

 

c) -√2(3-√2) =

= -3*√2 -√2*(-√2) =

= -3√2 +2 =

= 2 -3√2

 

d) (7-√3)(7+√3)=

Stosuję wzór skróconego mnożenia : (a -b)(a +b) = a² -b²

= 7² - (√3)² =

= 49 -3 =

= 46

 

e) (2√6-3)(√6+4)=

= 2√6*√6 +4*2√6 -3*√6 -3*4 =

= 2*6 +8√6 -3√6 -12 =

= 5√6

 

f) (√7-2√3)(2√7+√3)=

= 2√7*√7 +√7*√3 -2√3*2√7 -2√3*√3 =

= 2*7 + √21 - 4√21 - 2*3 =

= 14  - 3√21 - 6 =

= 8 -3√21

 

g) (√2+√3)(3√2-√3)=

= 3√2*√2 -√2*√3 +3√2*√3 -√3*√3 =

= 3*2 - √6 + 3√6 - 3 =

= 6 +2√6 -3 =

= 3 +2√6

 

h(3-4√10)¹⁰=

= [(3 -4√10)² ]⁵ =

= [ 3² -2*3*4√10 +(4√10)²]⁵ =

= [ 9 - 24√10 + 16*10 ]⁵ =

= [169 - 24√10]⁵ =

= [( 169 -24√10 )²]² *(169 -24√10) =

= [(169²-2*169*24*√10 +( 24*√10)²]²*(169 - 24√10) =

= [ 28561 - 8112√10 + 576*10 ]²( 169 - 24√10) =

= [ 28561 + 5760 - 8112*√10]²*(169 -24√10) =

= [ 34321 - 8112√10)²*(169 -24√10) =

= [ 34321²-2*22801*8112*√10 + 8112²*10]*( 169 - 24√10) =

 

zbyt duży wynik do wymnożenia

 

(3-4√10)¹⁰ można to zapisać tak:

[(3 -4√10)²]²*[(3 -4√10)²]²*(3 -4√10)²