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*CHALLENGE.
1.SIMPLIFY : SQUARE ROOT OF 16^8+8^4/16^2+16^7

2.EXPRESS 1 + 1/1+SQUARE ROOT OF 2 - 1/1-SQUARE ROOT OF 2 IN THE FORM A+B SQUARE ROOT OF 2, WHERE A AND BARE RATIONAL NUMBERS. FIND A + B.

3. ARRANGE THE NUMBERS- SQUARE ROOT OF 6, 1+ SQUARE ROOT OF 2, 2 SQUARE ROOT OF 5/3 IN INCREASING ORDER WITHOUT THE USE OF CALCULATOR.

4. WHAT IS THE POSITIVE GEOMETRIC MEAN FOR THE POSITIVE DIVISORS OF 2012?

5. WHAT IS THE SIMPLEST FORM OF 12/SQUARE ROOT OF 2 + SQUARE ROOT OF 3 + SQUARE ROOT OF 5?

6. A CUBE IS INSCRIBED IN A SPHERE WHOSE DIAMETER IS 12 CM. WHAT IS THE VOLUME OF THE CUBE?

7. THE DIFFERENCE BETWEEN 2 POSITIVE NUMBERS IS 2 SQUARE ROOT OF 3 AND THE PRODUCT OF THE 2 NUMBERS IS SQUARE ROOT OF 2. FIND THE ABSOLUTE VALUE OF THE DIFFERENCE OF THEIR RECIPROCALS.

whew - hard to concentrate WITH ALL THE SHOUTING!!!

#1:
(16^8+8^4)/(16^2+16^7)
= (16^8+16^2)/(16^2+16^7)
= 16^2 (16^6+1) / 16^2 (1+16^5)
= (16^6+1)/(16^5+1)
don't see any way to simplify that, so you wind up with
√(16^6+1)/(16^5+1)
it will be very very close to √16=4

#2:
1/(1+√2) - 1/(1-√2)
put all over a common denominator
[(1-√2) - (1+√2)]/(1^2 - √2^2)
= -2√2/(1-2)
= 2√2

#3:
√6,1+√2,2√5/3
try squaring the values. The squares will be in the same order as the numbers:
6,3+2√2,20/9
3+3,3+2√2,3-7/9
Since √2 < 1.5, 3+2√2 < 3+3
Looks like they should be in order:
2√5/3, 1+√2, √6

#4:
2012 = 2^2 * 503
The divisors are thus
1,2,4,503,1006,2012
geometric mean is sixthroot(2012^3) = √2012 = 2√503

#5:
12/√2 + √3 + √5
= 6√2 + √3 + √5
Hmmm. not much sense
12/(√2+√3+√5) = 3√2 + 2√3 - √30
not sure that's any simpler

#6:
The main diagonal of the cube is the diameter of the sphere. So, the side s of the cube is

3s^2 = 12
s=2
volume = 8

#7:
x-y = 2√3
xy = √2
1/x - 1/y = (y-x)/xy = -2√3/√2 = -√6
so, |1/x - 1/y| = √6

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