I don't understand: Compare taking a root and raising a power. Support your observations with your own examples.

Question

This is Algebra II (for dummies - college level - only Pass or Fail, no credits given.)

Answer 1

In algebra pairs of operations exist that are called "inverse operations."

Some are addition and subtraction, multiplication and division. These pairs of operation "undo" each other, like standing up and sitting down, opening a door and closing a door.

 

Raising to a power - evolution - and taking a root - involution - are inverse operations.

 

4³ = 64, ³√64 = 4 

Answer 2

by "taking a root", I assume you mean taking a square root. 

 

We know that when you multiply the square root of a number by that same value, you get the number. √n * √n  =  n.

 

And we know that any value raised to the power of one is equal to that value.  v¹  =  v.

 

we also know that the product of value raised to given power and that value raised to another power is equal to the value raised to the sum of the two powers.  E.g.,   a² (= a*a) * a³ (= a*a*a)  =  a*a * a*a*a  =  a⁵.

 

So, if we can express taking the square root of a number as raising that number to a certain power, √n  =  n^x , what would that power be?

                                 √n * √n =  n^x * n^x  =  n^x+x=2x  =  n¹  =  n.  if  n^2x  =  n¹, then 2x must equal 1  (2x = 1).

When you solve for x, you get x  =  1/2.

 

In other words, taking a [square] root is like raising to a power that is a fraction.