Question

What is the principle of powers? Describe in your own words. When solving a radical equation, how do we use the principle of powers to eliminate radicals? Demonstrate with an example.

Answer 1

roots and powers are inverse operations, just like add/subtract and multiply/divide

2+6=8 can be written 2=8-6 or 6=8-2

2*6=12 can be written as 12/6-2 or 12/2=6

sqrt(9) = 9^(1/2)
[sqrt(9)]^2 = [9^(1/2)]^2
9 = 9^[1/2 * 2] = 9^1 = 9

Roots are fractional powers

x^(m/n) is the nth root of x^m
or, it's [x^(1/n)]^m

So, whenever you have a radical involved, to get rid of it, raise everything to the power of the index

If you have a power of n, take the nth root of everything to get rid of the exponent.

It's just like having 6x+3=9
to get rid of the 3, subtract from both sides; then to get rid of the 6, divide both sides by 6.

Powers and roots work just the same way.