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Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
5.1: Working with Radicals
Consider the number 25:
25 = 52
and
25 = (–5)2.
x 2  25
*Note:
25 

Consider a square with area of 10.
The side length is the principal square root of 10; that
is 10 . Since 10 is not a perfect square, 10 cannot be
simplified and it is left as a radical.
Convert Mixed Radicals to Entire Radicals
Example
Your Turn
Express each mixed radical in entire radical form. Identify the values of the
variable for which the radical represents a real number.
Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
Compare and Order Radicals
Example Order the following numbers from least to greatest without a calculator.
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 that can 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 perfect square
factor of 9, which can be removed. 18 = 9 2 = 9  2 = 3 2 .
Express Entire Radicals as Mixed Radicals
Example Convert each entire radical to a mixed radical in simplest form.
Your Turn
Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
Restrictions on Variables
If a radical represents a real number and has an even index, the radicand must
be non-negative. The radical 4  x has an even index. So, 4 – x must be greater
than or equal to zero.
Your Turn
Example State the restrictions on the following radical expressions.
a)
x  2 1
b)
x5
2
 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
mixed form before identifying like radicals.
Add and Subtract Radicals
Example Simplify radicals and combine like terms.
Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
Your Turn
Simplify radicals and combine like terms.
5.2: Multiplying and Dividing Radical Expressions
Objectives:
 Performing multiple operations on radical expressions
 Rationalizing the denominator
 Solving problems that involve radical expressions
Multiplying Radicals
When multiplying radicals, multiply the coefficients and multiply the radicands. You can only
multiply radicals if they have the same index.
Ex.
 2 7  4 75 
Radicals can be simplified before multiplying!
Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
Example 1) Multiply, and simplify the products where possible.
Your Turn
Multiply. Simplify where possible.
Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
Dividing Radicals
When dividing radicals, divide the coefficients and then divide the radicands. You
can only divide radicals that have the same index.
Rationalizing Denominators (We don’t want radicals in the final denominator!)
To simplify an expression that has a radical in the denominator, you need to
rationalize the denominator, see the example below.
For a binomial denominator that contains a square root, multiply both the
numerator and denominator by a conjugate of the denominator.
Example 2) Simplify each expression.
a)
24x 2
,x  0
b)
3x


11
57
Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
Your Turn
Simplify each quotient.
a)
2 51
b)
3
6
4x  1
,x  0


5.3: Radical Equations (Part I)
Radical Equations are equations with radical signs in them. As usual, we are trying to
isolate the variable and determine possible values for the unknown.
Ex 1
a) State the restrictions on x in 5  2x 1  12 if the radical is to be a real
number.
b) Solve


5  2x 1  12
Pre-Calculus 11
Example 2)
Chapters 5: Radical Expressions and Equations
Identify the restrictions on n in n  5  n  7 if the radical is to be a real
number. Then, solve the equation.

Your Turn
Identify any restrictions on m in m  2m  3  6 if the radical is a real number. Then,
solve the equation. Check your solution(s).

Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
To solve radical equations:
1.
2.
3.
4.
State any restrictions on the variables.
Isolate the radical & square both sides.
Solve the remaining quadratic equation.
Check your solution(s). Reject any extraneous roots.
*Recall: Extraneous roots are solutions that do not satisfy any initial conditions.
5.3: Radical Equations (Part II)
Example Solve 7  3x  5x  4  5 . Check your solution(s).

Pre-Calculus 11
Chapters 5: Radical Expressions and Equations
Your Turn
Example Solve 9  4 x  1  x  2 . Check your solution(s).
Example What is the speed, in m/s, of a 0.4-kg football that has 28.8 J of kinetic
1
energy? Use the kinetic energy formula, Ek = mv2, where Ek represents the
2
kinetic energy, in joules; m represents mass, in kilograms; and v represents
speed, in m/s.