Question #c2de7
3 Answers
Explanation:
#"note that "•color(white)(x)sqrtaxxsqrta=(sqrta)^2=a#
#x^2=(sqrt(sqrt5+2)+sqrt(sqrt5-2))^2/(sqrt(sqrt5+1))^2#
#"expand numerator/denominator using FOIL method"#
#x^2=(sqrt5+2+2(sqrt5+2)(sqrt5-2)+sqrt5-2)/(sqrt5+1)#
#color(white)(x^2)=(2sqrt5+2(5-4))/(sqrt5+1)=(2sqrt5+2)/(sqrt5+1)#
#"rationalise the denominator"#
#"multiply numerator/denominator by "(sqrt5-1)#
#((2sqrt5+2)(sqrt5-1))/((sqrt5+1)(sqrt5-1)#
#"expand numerator/denominator using FOIL"#
#=(10-2)/(5-1)=8/4=2#
#rArrx^2=2#
Explanation:
#"note that " sqrtaxxsqrtb=sqrt(ab)#
#"simplify each fraction by rationalising it's denominator"#
#color(blue)"fraction 1"#
#((7sqrt3)(sqrt10-sqrt3))/((sqrt10+sqrt3)(sqrt10-sqrt3))#
#=(7sqrt30-21)/(10-3)#
#=(7sqrt30-21)/7=sqrt30-3to(color(red)(1))#
#color(blue)"fraction 2"#
#((2sqrt5)(sqrt6-sqrt5))/((sqrt6+sqrt5)(sqrt6-sqrt5))#
#=(2sqrt30-10)/(6-5)#
#=2sqrt30-10to(color(red)(2))#
#color(blue)"fraction 3"#
#((3sqrt2)(sqrt15-3sqrt2))/((sqrt15+3sqrt2)(sqrt15-3sqrt2))#
#=(3sqrt30-18)/(15-18)#
#=(3sqrt30-18)/(-3)#
#=-sqrt30+6to(color(red)(3))#
#"put all this together gives"#
#sqrt30-3-(2sqrt30-10)-(-sqrt30+6)#
#=cancel(sqrt30)-3cancel(-2sqrt30)+10cancel(+sqrt30)-6#
#=1#
Explanation:
#"begin with simplifying the denominator"#
#"note the following"#
#•color(white)(x)sqrt(ab)hArrsqrtaxxsqrtb#
#"for example"#
#sqrt80=sqrt(16xx5)=sqrt16xxsqrt5=4sqrt5#
#rArrsqrt10+sqrt20+sqrt40-sqrt5-sqrt80#
#=sqrt10+2sqrt5+2sqrt10-sqrt5-4sqrt5#
#"collecting like radicals gives"#
#=3sqrt10-3sqrt5=3(sqrt10-sqrt5)#
#rArr15/(sqrt10+sqrt20+sqrt40-sqrt5-sqrt80)#
#=cancel(15)^5/(cancel(3)^1(sqrt10-sqrt5))=5/(sqrt10-sqrt5)#
#"multiply numerator/denominator by conjugate surd"#
#=(5(sqrt10+sqrt5))/((sqrt10-sqrt5)(sqrt10+sqrt5))#
#=(5sqrt10+5sqrt5)/(10-5)=(5sqrt10+5sqrt5)/5#
#=sqrt10+sqrt5#
