please simplify

Question

a. √162

b. 5√(2) - 2 √5 +√125 - √8

c. √3(4√(6)+2√3)

d. √2(2√2=2) -3(5√2+1)

 

Answer 1

a) √(162)

We break 162 down into the factors of 81 and 2.

= √(81*2)

= √(81)√2

= 9√2

√(162) = 9√2.

Answer 2

b) The radical expression is 5√2 - 2√5 + √125 - √8

= 5√2 - 2√5 + √(5*25) - √(4*2)

{Apply the radical property √(ab) = (√a √b) where a, b ≥ 0 }

= 5√2 - 2√5 + √5√25 - √4√2

= 5√2 - 2√5 + √5(5) - 2√2

= 5√2 - 2√2 - 2√5 + 5√5

= 3√2 + 3√5

5√2 - 2√5 + √125 - √8 = 3√2 + 3√5.

Answer 3

c) The radical expression is √3(4√(6)+ 2√3)

= √3(4√(3*2)+ 2√3)

= √3(4√3√2)+ 2√3)

= √3(4√3√2) + √3(2√3)

= √3√3(4)√2) + √3√3(2)

{ Apply the radical property (√a √a) = a where a ≥ 0}

= 3(4)√2 + 3(2)

= 12√2 + 6

√3(4√(6)+ 2√3) = 12√2 + 6.

Answer 4

d) The radical expression is √2(2√2+2) - 3(5√2 + 1)

= (√2)(2√2) + (√2)(2) - (3)(5√2) + (- 3)(1)

= 2(√2)(√2) + 2√2 - 15√2 - 3

{ Apply the radical property (√a √a) = a where a ≥ 0}

= 2(2) - 13√2 -  3

= 4 - 13√2 -  3

= 1 - 13√2

√2(2√2+2) - 3(5√2 + 1) = 1 - 13√2.