9.3 Simplifying Radicals

Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b2 = a. In order to find a square root of a, you need a # that, when squared, equals a.

If x2 = y then x is a square root of y. In the expression , is the radical sign and 64 is the radicand. 1. Find the square root: 8 or -8

3. Find the square root: 11, -11 4. Find the square root: 21 or -21

6. Use a calculator to find each square root 6. Use a calculator to find each square root. Round the decimal answer to the nearest hundredth. 6.82, -6.82

What numbers are perfect squares? 1 • 1 = 1 2 • 2 = 4 3 • 3 = 9 4 • 4 = 16 5 • 5 = 25 6 • 6 = 36 49, 64, 81, 100, 121, 144, ...

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

Product Rule for Radicals If and are real numbers,

Simplifying Radicals Example Simplify the following radical expressions. No perfect square factor, so the radical is already simplified.

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

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

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

Find a perfect square that goes into 147. 1. Simplify Find a perfect square that goes into 147.

Find a perfect square that goes into 605. 2. Simplify Find a perfect square that goes into 605.

Simplify .

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

6. Simplify Multiply the radicals.

Multiply the coefficients and radicals. 7. Simplify Multiply the coefficients and radicals.

Multiply and then simplify

How do you know when a radical problem is done? No radicals can be simplified. Example: There are no fractions in the radical. Example: There are no radicals in the denominator. Example:

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

That was easy!

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. 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.

There is a radical in the denominator! 8. Simplify. Divide the radicals. Uh oh… There is a radical in the denominator! Whew! It simplified!

9. Simplify Uh oh… Another radical in the denominator! Whew! It simplified again! I hope they all are like this!

10. Simplify Since the fraction doesn’t reduce, split the radical up. Uh oh… There is a fraction in the radical! 10. Simplify Since the fraction doesn’t reduce, split the radical up. How do I get rid of the radical in the denominator? Multiply by the “fancy one” to make the denominator a perfect square!