## Presentation on theme: "The Radical Square Root"— Presentation transcript:

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 = 23 = 33 = 43 = 53 = 63 = 73 = (x4)3 = x12 (-2y2)3 = -8y6

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)

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: