+128475 First one.....convert to exponential form
5 ( 4n)^(1/4)
__________
2 (27n)^(1/4)
We can "cancel" the n's and get
5 (4)^(1/4)
_________ multiply top/bottom by (27)^(3/4)
2 (27)^(1/4)
5 ( 4 )^(1/4) (27)^(3/4)
__________________
2 (27)^(1/4) 27^(3/4)
5 [ 4 (27)^3 ] ^(1/4)
_________________
2 * 27
5 [ 4 (3^3)^3 ] ^(1/4)
________________
54
5 [ 4 * 3^9 ]^(1/4)
______________
54
Note : (3^9)^(1/4) = 3^(9/4) = 3^(2 + 1/4)...we can take 3^2 out of the radical and leave one power of 3 inside
5 * 3^2 [ 4 * 3 ]^(1/4)
_________________
54
5 * 9 [ 12]^(1/4)
_____________
54
5 [12]^(1/4)
_________
6
5 4√12
______
6
+2448 Thanks for helping! I still don't quite understand though, there are a lot of steps and it's confusing >.<
+128475 Yeah I know.....we are using a lot of exponential rules here
Here are a few that I employed
m √ [ a^n ] converts to exponential form (a)^(n/m)
And
a^(m) * a^(n) = a^(m + n)
And
Remember.... we can take something out of a radical when we have (a)^(n/m) and n ≥ m
For instance...let's suppose that we have 3 √ [ 4^5]
In exponential form....we have (4)^(5/3)
To determine how many powers of 4 come out of the radical and how many stay in.....write the exponent as a mixed number
4^(1 +2/3)
The whole number tells us that 1 power of 4 comes out of the radical and the numerator of the fraction tells us that 2 powers of 4 stay inside the radical...so we have
4 3√ [4^2 ] = 4 3√ [ 16 ]
But note that 16 = 2^4....so...we can apply this rule once more
We have 4 3√ [2^4 ] = 4 (2)^(4/3)
To determine how many powers of 2 that we can take out of the radical and how many we can leave in...write this as
4 (2)^(1 + 1/3)....so....we can take 1 power of 2 out of the radical and 1 power of 2 stays in
So..putting this all together, we have
(4 * 2) 3√2 = 8 3√2
Hang in there, RP......you will begin to understand this
+128475 The second one is easier, RP
√ [5 p^3 ]
_________ = (1/3) √ [ (5 p^3) / (2p^3) ]
3 √ [2 p^3 ]
The p^3's will "cancel" and we are left with
(1/3) √ [ 5 / 2 ] =
√ 5
_____ (do you see this ??? )
3 √2
We want to get rid of the radical in the denominator [ this is called "rationalizing the denominator" ]
We can do this in this manner.....Multiply top/bottom by √2 and note that √2 *√2 = √4 = 2
√5 * √2
__________
3 √2 * √2
√10
_____
3 * 2
√10
_____
6
+128475 I let you in on a secret, RP....
In most of the higher math classes I've had....I've rarely encountered these types of radicals.....these are mostly " textbook" exercises.....once you have to suffer through this.....you probably won't see this stuff, again !!!!!
[ Something to look forward to, huh ??? ]
