Simplifying Square Roots Square Roots: The square root of a number is one of its two equal factors. (Using symbols: a is the square root of b if a2 = b.) Example: The square root of 36 is 6 since 66 = 36. The positive square root is called the principle square root. We will mainly be concerned with the principle square root. Note: Negative real numbers do not have square roots because any nonzero real number is positive when squared. (No number multiplied by itself will give a negative real number.) The number under the radical symbol is called the radicand. (49 and 81 are the radicands.)
Example 1. Simplify the following radical expressions. Answers: Your Turn Example #2 Simplify the following radical expressions.
Writing Radical Expressions in Simplest Radical Form: Write the square root as a product of two square roots where one of the radicands is the largest perfect square that divides evenly into the original number. Then replace the square root with the whole number it is equal to. Leave as multiplication. Note: examples of perfect squares are 1, 4, 9, 16, 25, 36, 49, etc.
The factors of 40 are: The largest perfect square is 4. So we will rewrite the square root using the 4 and 10. 3. Now replace the square root of 4 with 2 and we’re done. The factors of 72 are: The largest perfect square is 36. So we will rewrite the square root using the 36 and 2. Now replace the square root of 36 with 6 and we’re done. Your Turn Problem #4 Simplify the following radical expressions.
Simplifying Square Roots that Involve Fractions We will now need the following property: In general, Property for Simplifying Radical Expressions that Involve Quotients.
5. Separate into the square root of the numerator divided by the square root of the denominator. Then simplify each (write both in simplest radical form). Separate into the square root of the numerator divided by the square root of the denominator. Then simplify each (write both in simplest radical form). Your Turn Example #6 Simplify the following radical expressions. Answers:
Your Turn Example #6 Simplify the following radical expressions. Answers:
Rationalizing the Denominator (Square Roots) In the last example, the denominators were perfect square roots. The numerator still contained a radical but not the denominator. A rational expression (a fraction) is not considered simplified if it contains a radical in the denominator. The process of “rationalizing the denominator” will take care of this. Rationalizing the Denominator (Square Roots) Observe the following: If a square root is multiplied by itself, the result is the radicand (without square root). Procedure: Rationalizing the denominator of a square root. (If the denominator contains a non-perfect square root) 2. Then simplify.
Separate into the square root of the numerator divided by the square root of the denominator. Then multiply the denominator by itself and multiply the numerator by the same number. Answers:
Simplify the following radical expressions. Your Turn Example #8 Simplify the following radical expressions. Answers: