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Square Roots and Radical Expressions
Solving Radical Equations
Simplifying Radical Expressions
Irrational Numbers in General and Square Roots in Particular
Roots of Polynomials
Simplifying Radical Expressions
Exponents and Radicals
Products and Quotients Involving Radicals
Roots of Quadratic Equations
Radical Expressions
Radicals and Rational Exponents
Find Square Roots and Compare Real Numbers
Radicals
Radicals and Rational Exponents
Theorems on the Roots of Polynomial Equations
SYNTHETIC DIVISION AND BOUNDS ON ROOTS
Simplifying Radical Expressions
Exponents and Radicals
Properties of Exponents and Square Roots
Solving Radical Equations
Rational Exponents and Radicals,Rationalizing Denominators
Rational Exponents and Radicals,Rationalizing Denominators
Quadratic Roots
Exponents and Roots
Multiplying Radical Expressions
Exponents and Radicals
Solving Radical Equations
Solving Quadratic Equations by Factoring and Extracting Roots
Newton's Method for Finding Roots
Roots of Quadratic Equations Studio
Roots, Radicals, and Root Functions
Review division factoring and Root Finding
Radicals
Simplifying Radical Expressions
Multiplying and Simplifying Radical Expressions
LIKE RADICALS
Multiplication and Division of Radicals
Radical Equations
BOUNDING ROOTS OF POLYNOMIALS
   

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!

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