The Radical Square Root The square root of any real number is a number, rational or irrational, that when multiplied by itself will result in a product that is the original number The Radical Radical sign Square Root Radicand Every positive radicand has a positive and negative sq. root. The principal Sq. Root of a number is the positive sq. root. A rational number can have a rational or irrational sq. rt. An irrational number can only have an irrational root.

=  7.9 =  232 225 = +15 529 = +23 Model Problems Find to the nearest tenth: = 13.4 =  7.9 =  11.4 =  232 =  64.4 Find the principal Square Root: 225 = +15 529 = +23 Simplify: = |x| = x = 2x8 = x + 1

Index of 2 Square Root Index of 2 radical sign radicand index of a number is one of the two equal factors whose product is that number Square Root Index of 2 has an index of 2 Every positive real number has two square roots The principal square root of a positive number k is its positive square root, . If k < 0, is an imaginary number

Index of 3 Cube Root Index = 3 radical sign radicand index of a number is one of the three equal factors whose product is that number has an index of 3 principal cube roots

nth Root The nth root of a number (where n is any counting number) is one of n equal factors whose product is that number. k is the radicand n is the index is the principal nth root of k 25 = 32 (-2)5 = -32 54 = 625

Index of n nth Root Index of n radical sign radicand index of a number is one of n equal factors whose product is that number nth Root Index of n has an index where n is any counting number principal odd roots principal even roots

Radical Rules! True or False: T T T

In general, for non-negative numbers a, b and n Radical Rule #1 In general, for non-negative numbers a, b and n Example: simplified = x4 = x3 Hint: will the index divide evenly into the exponent of radicand term?

In general, for non-negative numbers a, b, and n Radical Rule #2 True or False: If and T T Transitive Property of Equality If a = b, and b = c, then a = c then In general, for non-negative numbers a, b, and n Example:

Perfect Squares – Index 2 12 144 11 121 100 10 9 81 8 64 7 49 6 36 5 25 4 16 3 9 4 2 1

Perfect Square Factors Find as many combinations of 2 factors whose product is 75 Factors that are Perfect Squares Find as many combinations of 2 factors whose product is 128

Find as many combinations of 2 factors whose product is 80 Simplifying Radicals Simplify: answer must be in radical form. Find as many combinations of 2 factors whose product is 80 perfect square comes out from under the radical To simplify a radical find, if possible, 2 factors of the radicand, one of which is the largest perfect square of the radicand. The square root of the perfect square becomes a factor of the coefficient of the radical.

Perfect Cubes 13 = 1 23 = 8 33 = 27 43 = 64 53 = 125 63 = 216 73 = 343 (x4)3 = x12 (-2y2)3 = -8y6

Simplifying Radicals Simplify: answer must be in radical form. 1) Factor the radicand so that the perfect power (cube) is a factor 2) Express the radical as the product of the roots of the factors 3) Simplify the radical containing the largest perfect power (cube)

Simplifying Radicals Simplify: 1) Change the radicand to an equivalent fraction whose denominator is a perfect power. 2) Express the radical as the quotient of two roots 3) Simplify the radical in the denominator

Simplify: Model Problems KEY: Find 2 factors - one of which is the largest perfect square possible

Model Problems Simplify: