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**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.

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**= 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

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**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

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**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

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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

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**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

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Radical Rules! True or False: T T T

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**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?

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**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:

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**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

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**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

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**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.

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Perfect Cubes 13 = 23 = 33 = 43 = 53 = 63 = 73 = (x4)3 = x12 (-2y2)3 = -8y6

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**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)

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**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

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**Simplify: Model Problems**

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

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Model Problems Simplify:

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(Using symbols: a is the square root of b if a2 = b.)

(Using symbols: a is the square root of b if a2 = b.)

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