-PLEASE INCLUDE SOLUTIONS AND EXPLAIN HOW YOU DID IT PLEASE.
*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.
PLEASE ANSWER AS SOON AS POSSIBLE. PLEASEEEEE. WITH SOLUTIONS . HOW YOU DID IT PLEASEEEEEEEEE.
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
How about
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