Simplifying Radicals

Perfect Squares 64 225 1 81 256 4 100 289 9 121 16 324 144 25 400 169 36 196 49

Simplify = 2 = 4 = 5 This is a piece of cake! = 10 = 12

Simplify Using Perfect Square Method Find a perfect square that goes into 147

Find a perfect square that goes into 605 Simplify Find a perfect square that goes into 605

Simplify = = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM = = = = = =

Simplifying Radicals by Tree Method. Simplify the following. Step one, break the number down into a factor tree. Step two: Circle any pairs. The pairs bust out of the klink. 2 10 5 2

So… which is really just… so it ends up like this… This guy didn’t have a partner and so is left in prison This guy gets away This guy gets caught and is never heard from again But teacher, teacher, what happens to the other guy? He’s Dead, gone, went bye bye, won’t see him again! The guy that got away The guy left in prison

Another Example Simplify the following: Step 1: Make a factor tree So that leaves: Which is: 14 10 2 7 2 5

Cake Method Video on Cake Method

Your Turn! Simplify the following: 1. 2. 3. 4. 8

Simplify = = LEAVE IN RADICAL FORM = = =

Simplify = = LEAVE IN RADICAL FORM = = =

Adding or Subtracting Radicals To add or subtract square roots you must have like radicands (the number under the radical). Then you add or subtract the coefficients! Sometimes you must simplify first:

Example 1:

Try These

DO NOW

Multiplying Radicals * To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals.

Radical Product Property ONLY when a≥0 and b≥0 For Example: Equal

Multiplying Radicals 1. Multiply terms outside the radical together. 2. Multiply terms inside the radical together. 3. Simplify.

Multiplication and Radicals Simplify the expression:

Multiply then simplify

You can multiply using distributive property and FOIL. Multiplying Radicals You can multiply using distributive property and FOIL.

Multiply: You try.

Using the Conjugate to Simplify Expression Conjugate Product The radical “goes away” every time

_______ _______ To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator

Radical Quotient Property ONLY when a≥0 and b≥0 For Example: Equal

That was easy!

Fractions and Radicals Simplify the expressions: There is nothing to simplify because the square root is simplified and every term in the fraction can not be divided by 10. Make sure to simplify the fraction.

Dividing to Simplify Radicals No radicals in the denominator allowed Denominators must be “rationalized.” Multiply by 1in the form of √ √ 3 1

Using the Conjugate to Simplify Expression Conjugate Product The radical “goes away” every time

Dividing to Simplify Radicals conjugate Multiply by 1in the form of conjugate 5 1 11

Simplify: You try.

42 cannot be simplified, so we are finished. This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 42 cannot be simplified, so we are finished.

This can be divided which leaves the radical in the denominator This can be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator.

This cannot be divided which leaves the radical in the denominator This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. Reduce the fraction.

Summary: COMBINE LIKE TERMS “Outside” NUMBERS x NUMBERS To ADD and SUBTRACT COMBINE LIKE TERMS To MULTIPLY “Outside” NUMBERS x NUMBERS “Inside” NUMBERS x NUMBERS DISTRIBUTE and FOIL To DIVIDE “Rationalize” denominator using 1 Use conjugate ALWAYS SIMPLIFY AT THE END IF YOU CAN