Radicals

Radicals We say the square root of a or root a. The symbol is a radical. The positive number a is the radicand. We say the square root of a or root a.

The square root of a number n is the number which, when squared is n. For example : 5 and -5 are both the square root of 25, since 52 = 25 and (-5)2 = 25. √ 25 = 5 and -5

- The symbols: indicates the positive root indicates the negative root indicates both roots

Some examples: 8 – 6 ± 2

Why can’t we find a square root for -36... in other words can b2= -36? 62 = 36 (-6)2 = 36 6(-6) = -36 The square of a number can never be negative. Therefore, the square root of a negative number does not exist in the real numbers.

Radical Rules:

Product Rule-- The square root of a product is equal to the the product of the square roots.

Quotient Rule-- The square root of a quotient is equal to the quotient of the two radicals.

Important Products: n is positive = n

Radicals are simplified when: 1) the radicand has no perfect square factors 2) the denominator of a fraction is never under a radical. The product and quotient rules allow radical expressions to be simplified.

#1 “Take out” perfect square factors Rewrite the radicand as a product of its factors; with the largest perfect square factor possible. Use the product rule to simplify the root of the perfect square as a rational number, leaving the other factor under the radical.

#2 Rationalize the denominator: To create a fraction with a rational denominator, multiply both numerator and denominator by the the irrational number found in the denominator of the fraction.

Comparison with variables Examples Comparison with variables