pre-calculus 11 chapter 5 radical expressions and...
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Pre-Calculus 11 Chapter 5 Radical Expressions and Equations.
Date:__________________ Block:_____ Name:__________________
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: