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Introduction to Radicals

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If b 2 = a, then b is a square root of a. MeaningPositive Square Root Negative Square Root The positive and negative square roots Symbol Example

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

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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 principal square root of 9 is 3. negative square root: the negative square root of 9 is –3 and is shown like radical: the symbol which is read “the square root of a” is called a radical. radicand: the number or expression inside a radical symbol --- 3 is the radicand. perfect square: a number that is the square of an integer. 1, 4, 9, 16, 25, 36, etc… are perfect squares.

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Square Roots If a is a positive number, then is the positive (principal) square root of a and is the negative square root of a. A square root of any positive number has two roots – one is positive and the other is negative. Examples: non-real #

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

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

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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,...

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

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= 2 = 4 = 5 = 10 = 12

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Simplifying Radicals

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Simplifying Radical Expressions

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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.

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

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

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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

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Simplify: Square root of a variable to an even power = the variable to one-half the power.

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Simplify: Square root of a variable to an even power = the variable to one-half the power.

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Simplify:

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Simplify 1.. 2.. 3.. 4..

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Simplify 1.3x 6 2.3x 18 3.9x 6 4.9x 18

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+ To combine radicals: combine the coefficients of like radicals

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Simplify each expression

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Simplify each expression: Simplify each radical first and then combine.

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

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Simplify each expression

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Homework radicals 1 Complete problems 1-24 EVEN from worksheet

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

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Multiply and then simplify

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

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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.

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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.

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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.

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= X = Y 3 = P 2 X 3 Y = 2X 2 Y = 5C 4 D 5

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= = = =

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Homework: worksheet --- Non-Perfect Squares (#1-12) Classwork: Packet in Yellow Folder under the desk --- 2 nd page

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Homework radicals 2 Complete problems 1-15 from worksheet.

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