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Rational Exponents and Radicals Definitions

Defintion 1: If x ≠ 0, then and is undefined

Defintion 2: If x ≠ 0, then for any rational number n
For any positive integer n and m

Defintion 3: If n is odd, then “nth root of a” is such that
If n is even and a > 0, then “principal nth root of a” is such that

Defintion 4: If is real, then

So, for odd n: such that for even n and a > 0: such that
So, for odd n: for even n: for any a.

is	not a real number	if n is even and a < 0
	a rational number	if a is a perfect nth power (e.g. 9 = 3² is a perfect square)
	an irrational number	if a is not a perfect nth power

Defintion 5: If is real, then

Defintion 6: If is real, then

From now on r and s are any rational numbers, m, n and k are any intergers.

Rules

Rule 1:

Rule 2:

Rule 3:

Rule 4: For

Rule 5: For

If are real then Rule 6:

If all indicated roots are real then

Rule 8:

Sumplifying Radicals - Rationalizing denominator (square roots only!)

Simplifying Radicals - We will simplify in each step as example.

Step 1: Rewrite the expression with fractional exponents/powers.

By Rules 2, 3 and 5

Step 2: Reduce fractional powers

power = 2/3 - already reduced

power = 4/3 - already reduced

Step 3: If fractional power > 1, write it as a mixed fraction, follow example steps.

power = 2 - not fractional power

power = 2/3 < 1

power =

Step 4: Put things back in the main problem

Step 5: Pull out factors with integer powers

Step 6: Rationalize the denomenator of the 2nd factor

Denomenator , we need to make it , so multiply numerator by

[We get by solving for z]

Step 7: Pull out the common denomenator of the fractional power as

Step 8: Switch back to radical notation

—– Now it is fully simplified!