Lesson Examples of Radicals

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1. 3/sqrt(5) = 3*sqrt(5)/sqrt(5)^2 = (3*sqrt(5))/5

2. 5x/(sqrt(x)-sqrt(y)) = (5x/(sqrt(x)-sqrt(y)))*((sqrt(x)+sqrt(y))/(sqrt(x)+sqrt(y))) = 5x*(sqrt(x)+sqrt(y))/(x-y)

3.5*sqrt(5)+3*sqrt(45) = 5*sqrt(5)+3*sqrt(9*5) = 5*sqrt(5)+3*3*sqrt(5) = (5+9)*sqrt(5) = 14*sqrt(5)

4. 12*sqrt(18)+3*sqrt(50)+6*sqrt(20)+8*sqrt(45)-6*sqrt(147)
= 12*sqrt(3^2*2)+3*sqrt(5^2*2)+6*sqrt(2^2*5)+8*sqrt(3^2*5)-6*sqrt(7^2*3)
= 12*3*sqrt(2)+3*5*sqrt(2)+6*2*sqrt(5)+8*3*sqrt(5)-6*7*sqrt(3)
= 36*sqrt(2)+15*sqrt(2)+12*sqrt(5)+24*sqrt(5)-42*sqrt(3)
= 51*sqrt(2)+36*sqrt(5)-42*sqrt(3)

5. 3*root(3,5) * 4*root(4,3)
The LCM of 3 and 4 is 12.
root(3,5)=root(12,5^4)=root(12,625)
root(4,3)=root(12,3^3)=root(12,27)
3*root(3,5) * 4*root(4,3) = 3*root(12,625)* 4*root(12,27) = 12*root(12,625*27) = 12*root(12, 16875)

6. (7*root(3,2))/(4*root(5,3))
=(7*root(15,2^5))/(4*root(15,3^3))
=(7*root(15,32))/(4*root(15,27))
=(7/4)*root(15,32/27)

7. Which is greater, sqrt(1/3) or root(3, 1/2)?
Ans. sqrt(1/3) = root(6,(1/3)^3) = root(6,1/27)
= root(3,1/2)=root(6,(1/2)^2) = root(6,1/4)
Since root(6,1/4) > root(6,1/27), thus root(3, 1/2) is greater than sqrt(1/3).

8. (sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2)) = ((sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2)))*((sqrt(3)+sqrt(2))/(sqrt(3)+sqrt(2)))
=(sqrt(3)+sqrt(2))^2/(3-2) = (3+2*sqrt(3)*sqrt(2)+2) = 5+2*sqrt(6)

9. 1/(sqrt(3)-sqrt(2)-1) = (1/(sqrt(3)-(sqrt(2)+1)))*((sqrt(3)+(sqrt(2)+1))/(sqrt(3)+(sqrt(2)+1)))
=(sqrt(3)+(sqrt(2)+1))/(3-(sqrt(2)+1)^2)
=(sqrt(3)+(sqrt(2)+1))/(3-(2+1+2*sqrt(2)))
=(sqrt(3)+(sqrt(2)+1))/(-2*sqrt(2))
=(-(sqrt(3)+(sqrt(2)+1))/(2*sqrt(2)))*(sqrt(2)/sqrt(2))
=(-(sqrt(3)+sqrt(2)+1)*sqrt(2))/(4)
=-(sqrt(6)+2+sqrt(2))/4
=-(sqrt(6)+sqrt(2)+2)/4
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