how do u simplify expression using the properties of radicals and rational exponents

Question

like 4th root 192 over 4th root 6  or (-25)5/2

 

Answer 1

for fourth root 192 over fourth root 6... they are both fourth roots so you can divide 192 by 6.  Then you get fourth root 32.  Then ask, "what number times itself four times will go into 32?  the answer is 2 (2^4=16). So fourth root 16 times fourth root 2.  Fourth root of 16=2.  Final answer is 2 times fourth root 2.

(-25)^(5/2) is equivallent to asking what is the square root of (-25)^5?  In this case it is easier to start with the square root. (assuming you are familiar with imaginary numbers) Then sqrroot of (-25) is 5i, now take 5i to the 5th power.  5^5=3125 and i^5=i so final answer is 3125i.

Comments

its -25 to the power of 5/2 sorry

and how do u figure this out im trying to get it into my head but i cant is their any steps..? or special trickss..?

Answer 2

k, (meaning the negative is completed after?)  Unfortunately there are no tricks

Think of it this way, 5/2 = 5(1/2). a power of 1/2 is equivalent to a square root.  You can accomplish the exponent in any order.  Again, taking the square root of 25 first then taking 5 to the 5 power (=3125) is easier then taking 25 to the 5 and then taking the square root of it (assuming no calculator).  Then multiply by -1.  So final answer -3125