SOLUTION: Can anybody show me how to solve these problems: 1: Fifth root of -32s^15t^10 2: Fourth root of 256y^8 3: (2-sqrt5)*(2+sqrt5) I am so confused :( Thank you

Algebra ->  Radicals -> SOLUTION: Can anybody show me how to solve these problems: 1: Fifth root of -32s^15t^10 2: Fourth root of 256y^8 3: (2-sqrt5)*(2+sqrt5) I am so confused :( Thank you      Log On


   

Question 399360: Can anybody show me how to solve these problems:
1: Fifth root of -32s^15t^10
2: Fourth root of 256y^8
3: (2-sqrt5)*(2+sqrt5)
I am so confused :(
Thank you
Found 3 solutions by ewatrrr, josmiceli, stanbon:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

1: root%285%2C%28+-32s%5E15t%5E10%29%29+=+-2s%5E3t%5E2

2:root%284%2C%28+256y%5E8%29%29+=+4y%5E2

3: (2-sqrt5)*(2+sqrt5) = 4-5 = -1



Rules of Exponents:

I.  x%5Em%2Ax%5En+=+x%5E%28m%2Bn%29+   

II.  %28a%5Ep%29%5Eq+=+a%5E%28p%2Aq%29

III. root%28a%2Cp%29%2Aroot%28a%2Cq%29+=+root%28a%2Cp%2Aq%29

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
What you have to keep in mind is that, for example,
root%285%2C3200000%29+=+root%285%2C32%2A10%5E5%29
root%285%2C32%2A10%5E5%29+=+root%285%2C32%29%2Aroot%285%2C10%5E5%29
root%285%2C32%29%2Aroot%285%2C10%5E5%29+=+2%2A10
2%2A10+=+20
In reverse:
2%5E5+%2A+10%5E5+=+32%2A100000
----------------------------
Your problems:
root%285%2C-32s%5E15%2At%5E10%29+=+-2%2As%5E3%2At%5E2 answer
To prove it, solve in reverse:
%28-2%2As%5E3%2At%5E2%29%5E5+=+%28-2%29%5E5+%2A+%28s%5E3%29%5E5%2A%28t%5E2%29%5E5
Now, expanding this:

There are 5 (-) signs, so (-2)^5 is negative
%28-2%29%2A%28-2%29%2A%28-2%29%2A%28-2%29%2A%28-2%29+=+-32

s%5E%283%2B3%2B3%2B3%2B3%29+=+s%5E15
likewise:

+t%5E%282%2B2%2B2%2B2%2B2%29+=+t%5E10
So, raising the answer to the 5th power gives me
the thing that I took the 5th root of.
--------------------------------
(2)
root%284%2C+256y%5E8%29+=+root%284%2C256%29%2Aroot%284%2Cy%5E8%29
root%284%2C256%29%2Aroot%284%2Cy%5E8%29+=+4%2Aroot%284%2Cy%5E2%2Ay%5E2%2Ay%5E2%2Ay%5E2%29
4%2Aroot%284%2Cy%5E2%2Ay%5E2%2Ay%5E2%2Ay%5E2%29+=+4%2Ay%5E2 answer
Again, you can prove your answer if you do
%284%2Ay%5E2%29%5E4+=+256%2Ay%5E8
----------------------
(3)
%282+-+sqrt%285%29%29%2A%282+%2B+sqrt%285%29%29
First think generally about
%28a+%2B+b%29%2A%28a+-+b%29+=+a%5E2+%2B+ab+-+ab+-+b%5E2
a%5E2+-+b%5E2
Now just do a substitution:
a+=+2
b+=+sqrt%285%29
What is a%5E2+-+b%5E2 ?
2%5E2+-+%28sqrt%285%29%29%5E2
4+-+5
-1 answer
You could even check this with your calculator:
%282+-+2.2361%29%2A%282+%2B+2.2361%29
%28-.2361%29%2A%284.2361%29
-1.0001
close enough. The more figures I use for sqrt%285%29,
the closer I get to -1

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
1: Fifth root of -32s^15t^10
= -2s^3t^2
-----------------------------------
2: Fourth root of 256y^8
= 4y^2
-----------------------------------
3: (2-sqrt5)*(2+sqrt5)
= 4-5
= -1
======
Cheers,
Stan H.