SOLUTION: Divide. Rationalize the denominator if necessary. Then simplify each radical expression. 1) 18+ square root of 567 OVER 9 2) -9- square root of 108 OVER 3 3) 6- square roo

Algebra ->  Square-cubic-other-roots -> SOLUTION: Divide. Rationalize the denominator if necessary. Then simplify each radical expression. 1) 18+ square root of 567 OVER 9 2) -9- square root of 108 OVER 3 3) 6- square roo      Log On


   

Question 118065: Divide. Rationalize the denominator if necessary. Then simplify each radical expression.
1) 18+ square root of 567 OVER 9
2) -9- square root of 108 OVER 3
3) 6- square root of 20 OVER 2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first two to show you how to do these

#1

Start with the expression
%2818+%2B+sqrt%28567%29%29%2F9

First lets reduce sqrt%28567%29
-----------------------------------------------------
sqrt%28567%29 Start with the given expression
The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. This way the perfect square will become a rational number.
So let's list the factors of 567
Factors:
1, 3, 7, 9, 21, 27, 63, 81, 189, 567


Notice how 81 is the largest perfect square, so lets break 567 down into 81*7


sqrt%2881%2A7%29 Factor 567 into 81*7

sqrt%2881%29%2Asqrt%287%29 Break up the square roots using the identity sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29

9%2Asqrt%287%29 Take the square root of the perfect square 81 to get 9

So the expression

sqrt%28567%29

simplifies to

9%2Asqrt%287%29
-----------------------------------------------------

%2818+%2B+9%2Asqrt%287%29%29%2F9 Simplify the square root (using the technique above)

18%2F9+%2B+9%2Asqrt%287%29%2F9 Break up the fraction

2+%2B+9%2Asqrt%287%29%2F9 Divide 18%2F9 to get 2

2+%2B+1%2Asqrt%287%29 Divide 9%2F9 to get 1



So the expression
%2818+%2B+sqrt%28567%29%29%2F9

simplifies to

2+%2B+sqrt%287%29





#2




Start with the expression
%28-9+-+sqrt%28108%29%29%2F3

First lets reduce sqrt%28108%29
-----------------------------------------------------
sqrt%28108%29 Start with the given expression
The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. This way the perfect square will become a rational number.
So let's list the factors of 108
Factors:
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108


Notice how 36 is the largest perfect square, so lets break 108 down into 36*3


sqrt%2836%2A3%29 Factor 108 into 36*3

sqrt%2836%29%2Asqrt%283%29 Break up the square roots using the identity sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29

6%2Asqrt%283%29 Take the square root of the perfect square 36 to get 6

So the expression

sqrt%28108%29

simplifies to

6%2Asqrt%283%29
-----------------------------------------------------

%28-9+-+6%2Asqrt%283%29%29%2F3 Simplify the square root (using the technique above)

-9%2F3+-+6%2Asqrt%283%29%2F3 Break up the fraction

-3+-+6%2Asqrt%283%29%2F3 Divide -9%2F3 to get -3

-3+-+2%2Asqrt%283%29 Divide 6%2F3 to get 2



So the expression
%28-9+-+sqrt%28108%29%29%2F3

simplifies to

-3+-+2%2Asqrt%283%29