Introduction to Radicals

Introduction to Radicals

If b 2 a, then b is a square root of a. Meaning Positive Square Root Negative Square Root Example Symbol 9 3 The positive and negative square roots 9 3 9 3

Radical Expressions Finding a root of a number is the inverse operation of raising a number to a power. radical sign index n a radicand This symbol is the radical or the radical sign The expression under the radical sign is the radicand. The index defines the root to be taken.

square root: one of two equal factors of a given number. The radicand is like the “area” of a square and the simplified answer is the length of the side of the squares. Principal square root: the positive square root of a number; the 9 3 principal square root of 9 is 3. negative square root: the negative square root of 9 is –3 and is shown 9 3 like radical: the symbol radical. 3 which is read “the square root of a” is called a radicand: the number or expression inside a radical symbol is the radicand. perfect square: a number that is the square of an integer. 1, 4, 9, 16, 25, 36, etc are perfect squares. --- 3

Square Roots A square root of any positive number has two roots – one is positive and the other is negative. If a is a positive number, then a is the positive (principal) square root of a and a is the negative square root of a. Examples: 100 10 0.81 0.9 36 6 25 5 49 7 1 1 9 non-real #

What does the following symbol represent? The symbol represents the positive or principal root of a number. What is the radicand of the expression 4 5xy ? 5xy

What does the following symbol represent? The symbol represents the negative root of a number. What is the index of the expression 3 3 5x 2 y 5 ?

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, .

Perfect Squares 64 225 9 16 81 100 121 256 289 25 36 49 144 169 196 400 1 4 324 625

4 2 16 4 25 5 100 10 144 12

Simplifying Radicals

Simplifying Radical Expressions Product Property for Radicals ab a b 36 4 9 36 4 9 6 2 3 100 4 25 10 2 5

Simplifying Radical Expressions Product Property for Radicals 50 25 2 5 2 A radical has been simplified when its radicand contains no perfect square factors. Test to see if it can be divided by 4, then 9, then 25, then 49, etc. Sometimes factoring the radicand using the “tree” is helpful. 14 x x 7

LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 4*2 2 20 4*5 2 5 32 16 * 2 4 75 25 * 3 5 3 40 4 *10 2 10 2 2

LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 48 16 * 3 4 3 80 16 * 5 4 5 50 25 * 2 5 2 125 25 * 5 5 5 450 225 * 2 15 2

Steps to Simplify Radicals: 1. Try to divide the radicand into a perfect square for numbers 2. If there is an exponent make it even by using rules of exponents 3. Separate the factors to its own square root 4. Simplify

Simplify: x 12 2 6 x x 6 Square root of a variable to an even power the variable to one-half the

Simplify: y y 88 44 Square root of a variable to an even power the variable to one-half the

Simplify: 12 1 x x x 13 12 x x x 6 x

Simplify: 50 y 7 6 25 y 2 y 5y 3 2y

Simplify 1. 2 18 2. 3 8 3. 6 2 4. 36 2 . . . . 72

Simplify 1. 3x6 2. 3x18 3. 9x6 4. 9x18 9x 36

To combine radicals: combine the coefficients of like radicals

Simplify each expression 6 7 5 7 3 7 8 7 5 6 3 7 4 7 2 6 3 6 7 7

Simplify each expression: Simplify each radical first and then combine. 2 50 3 32 2 25 * 2 3 16 * 2 2 *5 2 3* 4 2 10 2 12 2 2 2

Simplify each expression: Simplify each radical first and then combine. 3 27 5 48 3 9 * 3 5 16 * 3 3*3 3 5* 4 3 9 3 20 3 29 3

LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 18 288 75 24 72

Simplify each expression 6 5 5 6 3 6 3 24 7 54 2 8 7 32

Simplify each expression 6 5 5 20 18 7 32 2 28 7 6 63

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

Multiply and then simplify 5 * 35 175 25 * 7 5 7 2 8 * 3 7 6 56 6 4 *14 6 * 2 14 12 14 2 5 * 4 20 8 100 8 *10 80

5 2 5* 5 25 5 7* 7 49 7 8* 8 64 8 x x* x x x 7 2 8 2 2 2

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

56 8 7 4*2 2 2

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. 6 7 6 * 7 42 49 7 7 42 7 42 cannot be simplified, so we are finished.

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. 5 10 1 * 2 2 2 2 2

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. 3 12 3 * 12 3 3 3 3 36 Reduce the fraction. 3 3 6 3 2

X Y 4 X 2 Y3 6 6 2 P X Y 4 4X Y 8 2 25C D 10 P2X3Y 2X2Y 5C4D5

X 3 X Y 5 2 X *X X Y 2 Y 4 Y Y