7.1 and 7.2 Simplifying/Multiplying/Dividing Radical expressions

Nth Roots

For any real number a and b, and any positive integer n, if an = b , then a is the nth root of b.

Real Number Examples: Find the roots: Square Root of 4 Square Root of -4 Cube Root of 8 Cube Root of -8 Fourth Root of 16 Fourth Root of -16 Fifth Root of 32 Fifth Root of -32

Variable Examples: Find the Square Root of: a1  2. a2 3. a3 4. a4

We always use the principal root when simplifying. When a number has two real roots, the positive root is called the principal root. We always use the principal root when simplifying.

Examples – Simplify : 1. 2.

3. 4.

5. 6.

7. 8. 9.

10. 11. 12.

If they are real numbers, then Multiplying and Dividing Radical Expressions If they are real numbers, then

Examples: 1. 2.

3. 4.

5. 6.

Dividing Radical Expressions

Examples: 1. 2.

Rationalizing the Denominator **Multiply the numerator and denominator by the denominator** Then Simplify Example: 1.

2. 3.

4. 5.

7.3 Adding, Subtracting, Multiplying and Dividing Binomial Radical Expressions

Adding Radical Expressions Use the same concept as that of adding or subtracting like variables. Example: 7 - 3x + 2x + 5 *Have to have like Terms to Add/Subtract*

Like Radicals are radical expressions that have the same index and the same radicand.

Like Radicals Unlike Radicals = =

Examples: 1. 2.

3. 4. 5. 6.

Always simplify radicals before combining! 1. 2.

3. 4. 5. 6.

Multiplying Radical Expressions When multiplying radicals, one must multiply the numbers OUTSIDE (O) the radicals AND then multiply the numbers INSIDE (I) the radicals.

Dividing Radical Expressions When dividing radicals, one must divide the numbers OUTSIDE (O) the radicals AND then divide the numbers INSIDE (I) the radicals. Remember to rationalize the denominator if needed!

Examples: 1. 2.

Multiplying Binomials To multiply, USE FOIL! Example 1:

2. 3.

Dividing Binomial Radicals To divide, Rationalize the denominator! (a + b)( a - b) = a2 – b2 These are called conjugates! They make radicals disappear!

Examples: 1.

2.

Solve: 1. 2.

3. 4.

Examples: 1.

2.

3.

4.

Practice: Practice 7-3 #1-30 Left Column Homework Practice: Practice 7-3 #1-30 Left Column

7.4 Rational Exponents

Homework Check

Rational Exponents

Rational Exponents are another way to write radicals.

Simplify each expression. 1. 2.

3.

4. 5.

6.

Converting to Radical Form 1. 2.

3. 4. 5.

Converting to Exponential Form 1. 2.

3. 4. 5.

Properties of Exponents also apply to Rational Exponents Properties of Exponents also apply to Rational Exponents! Write in Radical Form:

2. 3. 4. 5.

Simplify each expression. 1. 2.

3. 4.

Practice 7-4 # 1-25, 29-44 Left Column ONLY Homework Practice 7-4 # 1-25, 29-44 Left Column ONLY

7.5 Solving Radical Equations

GBMP Review y=0.5(2)^x 2.130% 3. 0.69 4. 32^(3/5)=8 5. x=43.3 6. ln(a^3/b^5) 7.x=0.1879 8. -2 9. X = 0.2693 10. $7708.03 11. $119.94 12. y = 250(1.60)^x

Homework Check

Radical Equations A radical equation is an equation that has a variable in a radicand or has a variable with a rational exponent. Are these Radical Equations?

We use inverse operations to solve equations. Solve: X2 = 4 The inverse of squaring a function is finding the square root.

Solve: X3 = 64 The inverse of cubing a function is finding the cube root.

The inverse of raising a function to the nth power is finding the nth root.

Solve the following. Check your solutions! 1. 2.

3. 4.

5.

6.

Solve (x)1/2 = 3 Recall that ½ = , so we square both sides! What do you know about 2 and ½? **To solve radical equations with rational exponents, raise each side to the reciprocal exponent!

……… … Therefore,

Examples: 1. 2.

3. 4.

Remember if the numerator is squared then we must do ± 1. 2.

3. 4.

The solutions are the x-intercepts! You may also solve radical equations by graphing! Set the equation equal to zero and graph. The solutions are the x-intercepts!

Given: The equation is already equal to zero Given: The equation is already equal to zero! y= Use the zero function to find the x-intercepts!