Algebra Tutorials!
Saturday 28th of May Home 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
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Expressions

Consider the radical expression Use the distributive property
to rewrite the expression in
factored form. Then simplify
the expression.

Like Terms & Combining Like Terms
Like terms
factors. To combine like terms means to add the
coefficients while leaving the variable and radical
factors unchanged.

Example 1
Perform the indicated operation.    Example 2
Perform the indicated operation.  Example 3
Perform the indicated operation.  If and are real numbers, then .
Specifically, if n = 2 and a = b, then Example 2
Perform the indicated operation. Example 3
Perform the indicated operation and simplify.  Example 4
Perform the indicated operation and simplify.  Example 4

Simplify i. Write the expression in
exponential form

ii. Perform the indicated
operation(s)

iii. Write the expression in radical
form

9.3 Rationalizing Denominators and Simplifying

denominator. The procedure for removing a radical from
the denominator is called rationalizing the denominator.
The product property of radicals is used to rationalize a
denominator.

If and are real numbers, then .
Specifically, if n = 2 and a = b, then Example 1
Rationalize a One-Term, Square Root
(n = 2) Denominator

1. Simplify (rationalize the denominator) 2. Simplify (rationalize the denominator) Example 2
Rationalize a One-Term, Cube
Root (n = 3) Denominator

 Note 1. Simplify (rationalize the denominator)  the goal is to make the radicand a perfect cube

2. Simplify (rationalize the denominator) 3. Simplify (rationalize the denominator) If and are real numbers, then

 Product Property Quotient Property  A radical expression is simplified if
1. There are no radicals in a denominator.
2. There are no fractions inside a radical symbol.
3. All radicands have no nth power factors.
4. The numerator and denominator of any rational
expression (fractions) have no common factors.

Example 3

1. Simplify 2. Simplify 3. Simplify Rationalize a Two-Term Denominator

Conjugate

The conjugate of the two-term expression a + b is a − b
and visa versa.

Example 4
For each of the following, identify the conjugate of the
expression. Then find the product of the expression and its
conjugate.

 Expression Conjugate Product a − b

Fact
The product of a square-root expression and it’s
conjugate is an expression containing no square
roots (i.e. a rational expression).

Example 5

Simplify Example 6

Simplify 9.5 Solve Square Root Equations

Recall that expressions are things we can be asked to
simplify, add, subtract, multiply, and divide. However,
equations (two equal expressions) are things we are asked
to solve. In this section we will solve square root
equations, such as, To Solve an Equation Containing One Square
Root Term

1. Isolate the square root term on one side of the equation.
2. Square both sides of the equation and solve.
3. Check the solution(s) in the original equation.

Example 1
1. Solve 2. Solve 3. Solve Watch for Extraneous Solutions
When both sides of an equation are squared it is possible
for the modified equation to have a solution that does not
satisfy the given equation - these false solutions are called
extraneous solutions and must be discarded.

Example 2

1. Solve 2. Solve 3. Solve Example 3 Solve each equation.   Example 4 Solve Example 5

1. Find the zeros & x-intercepts of 2. Find the y-intercept of f.

3. Verify the results by graphing f on your
calculator. 