Radicals Review 4 April 2011

Parts Coefficient Radical Sign Radicand – the number underneath the radical sign Radical Pronounced: 2 times the square root of 7 OR 2 radical 7

Simplest Radical Form When you cannot factor any more perfect squares from the radicand The radical cannot be simplified further We always want our answers to be in simplest radical form

Getting Radicals into Simplest Radical Form 1. Write the radicand as the product of factors, where one (or more) factors is a perfect square 2. Take the square root of any perfect squares (Remember to multiply any coefficients in front of the radical sign!) 3. Repeat until you cannot factor any more perfect squares from the radicand

Tips for Getting Radicals into Simplest Radical Form Always check if the radicand is perfect square! Check if factorable by common perfect squares – 4, 9, 16, or 25 If the radicand is prime (or if its only factors are prime), then it’s in simplest radical form Be persistent! You don’t have to find the largest perfect square the first time you factor the radicand

Examples

Your Turn: Write problems 1 – 8 in simplest radical form.

Adding and Subtracting Radicals You can only combine radicals with the same radicand (like radical terms)! 1. Rewrite all radicals in simplest radical form first! 2. Add or subtract the coefficients of like radical terms

Examples

Your Turn: For problems 9 − 12, simplify. Write the answer in simplest radical form.

Multiplying Radicals Multiply like parts coefficients * coefficients radicand * radicand Write answer in simplest radical form

Examples

Your Turn:

What is rationalizing? The process of algebraically removing a radical sign from one part of a fraction We generally rationalize the denominator (But we can rationalize the numerator.)

Why rationalize? The result is easier to estimate and understand Also shows up in solving limits (in calculus)

Definitions Monomial – An expression with exactly one term Example: 3y or –7x 3 Binomial – An expression with exactly two terms Example: 6 + 4x or 10y 4 – 8

Definitions, cont. Conjugates – binomial expressions, such as (a + b) and (a – b), that differ only in the sign of the second term Examples: (3 – x) and (3 + x) (4y 5 + 2x 2 ) and (4y 5 – 2x 2 ) The product of conjugates is a 2 – b 2

Definitions, cont. (3 + x) and (3 – x)

Definitions, cont. (4y 5 + 2x 2 ) and (4y 5 – 2x 2 )

Your Turn:

Rationalizing the Numerator Exact same process as rationalizing the denominator, except that we focus on the numerator instead of the denominator. Reappears in calculus

Examples