Pre-Calculus 11 Chapter 5 Radical Expressions and Equations.

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Radical equations can be used to model a variety of relationships—from

tracking storms to modeling the path of a football or a skier through the air. Radical

expressions and equations allow mathematicians and scientists to work more accurately with

numbers. This is important when dealing with large numbers or relations that are

sensitive to small adjustments. In this chapter, you will work with a variety of radical

expressions and equations including very large radicals as you analyse the cloud formations

on the surface of Saturn.

Lesson Notes 5.1: Working With Radicals

Objectives:

• converting between mixed radicals and entire radicals

• comparing and ordering radical expressions

• identifying restrictions on the values for a variable in a radical expression

• simplifying radical expressions using addition and subtraction

Consider the number 25 = 5 5 = 52 and 25 = (–5)(–5) = (–5)2.

So 25 has two square roots: 5 and –5. 5 is called the principal square root of 25; it

is written as 25 and represents the positive square root of 25, since 25 is a perfect square,

therefore 5 and 25 are equivalent.

Consider a square with area of 10.

The side length of the square is positive, so it is the

principal square root of 10; that is 10 . Since 10 is not a perfect square,

so 10 can not be simplified and it is left as a radical.

Like Radicals

Radicals with the same radicand and

index are called like radicals. When adding

and subtracting radicals, only like radicals

can be combined. You may need to convert

radicals to a different form (mixed or entire)

before identifying like radicals.

Restrictions on Variables

If a radical represents a real number and has an even index, the radicand must be

non-negative. The radical x4 has an even index. So, 4 – x must be greater than or equal

to zero.

Isolate the variable by applying algebraic

operations to both sides of the inequality symbol.

The radical x4 is only defined as a real number if x is less than or equal to

four. You can check this by substituting values for x that are greater than four, equal to four,

and less than four.

Convert Mixed Radicals to Entire Radicals

Example 1) Express each mixed radical in entire radical form. Identify the values of the variable for which

the radical represents a real number.

Your Turn

Radicals in Simplest Form

A radical is in simplest form if the following are true.

• The radicand does not contain a fraction or any factor which may be removed.

• The radical is not part of the denominator of a fraction.

For example, 18 is not in simplest form because 18 has a square factor of 9, which can be

removed. 18 = 29 = 233 is equivalent to the simplified form 3 2 .

Express Entire Radicals as Mixed Radicals

Example 2) Convert each entire radical to a mixed radical in simplest form.

Your Turn

Compare and Order Radicals

Example 3) Five bentwood boxes, each in the shape of a cube have the following diagonal lengths, in

centimetres. Order the diagonal lengths from least to greatest without using a calculator.

Your Turn

Order the following numbers from least to greatest: