calculate the following

Question

what is the fourth root of 16 how do you find it with no calculator

 

and 

 

what is the fifth root of -32 without a calculator 

Answer 1

Hi Deirdre!  What's great about these two problems is that solving the first solution can help you with the second solution.  It helps to know your basic squares.  The fourth root of 16 will be the base of a number to the 4th power, or the number n squared times n squared:

N4 = 16

N² X N² =16

You probably know the square root of 16.  If you are thinking 4, you are correct.

What number N², or N X N = 4?  If you are thinking 2, great!

Check the value for N:

N² X N² = 2² X 2², or 4 X 4 = 16.  So, 24 = 2 X 2 X 2 X 2=16

 

OK, now for the next equation N⁵ = -32

If you remember that any number times itself, negative or positive will = a positive number, then your 5th route number will have to be negative.

 You probably know that integers 2, 4, and 16 are multiples of 32.  Tip:  Any number ending in 2, 4, 6, 8 or multiple of 10, is a multiple of 2.  From your first equation, you know that 24 = 16.

 Then 2⁴ X 2¹ or 2⁴ X 2 = 2⁵.  Remember you can solve by adding exponents with the same root.

However you need a negative solution to get the answer -32 instead of 32.  So your n value must be negative 2.  N = -2

Check your results.  N^-4, or –N X –N X –N X –N= positive N⁴.  Multiply this positive number by a negativenumber and your answer will have the negative route.  N^-5 = -32, 2^-5 = -32.  I hope this helps!

Shirley

 

Answer 2

fourth root of 16

16=2*2*2*2=2⁴ (you should see that the answer is 2 from here, but...)

fourth root of 16=(2⁴)^(1/4)=(2)^(4*[1/4])=2

 

-32=(-2)(-2)(-2)(-2)(-2)=(-2)⁵

fifth root of -32=[(-2)⁵]^(1/5)=(-2)^(5*[1/5])=-2