Ch. 5 Notes - WordPress.com€¦ · Pre-Calculus 11 Date: _____ Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 1 of 29

Pre-Calculus 11 Date: _______________

Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 1 of 29

5.0 – Perfect squares and Perfect Cubes

A fast and efficient way to solve radicals is to recognize and know the ‘perfect’ numbers.

Perfect Squares Perfect Cubes

21 3122 3223 3324 3425 3526 3627 3728 3829 39

210 310211212 3

1213 32214 3

2215 33216 3

3217 34218 3

4219 35220 3

5221 3

6222 3

7223 3

8224 3

9225 3

10



5.1a – Working with Radical Numbers

Definition: Radical n xIndex

Radicand

Recap: Exponent Laws

2)( ba

2

b

a

x

3 x

Extension of Exponent Laws:

ba

b

a

In General: n ba = nn ba

nb

a=

n

n

b

a

Recall:Square roots – a number , is a r square root of a number x , if 2r x .

Note: A positive number always has two square roots, one positive and one negative, because:

It is impossible to obtain a negative number when a number is squared, therefore the square root of a negative number is NOT defined.

Notation: radical sign , denotes the positive square root only.

x means the positive square root of x , where 0x .



Example 1: Simplify. a) 2 500= b) – 0.16=

Cube Roots – A number , is a r cube root of a number x , if 3r x .

Note: The cube root of a positive number is _________________.

The cube root of a negative number is _________________.

3 x means the cube root of x

Example 2: Simplify. Round to 3 decimal places if necessary.

a) 3 64 b) 3 125 c ) 3 21 b) 3 125

note: is the same as

Higher Roots

An expression of the form n x is a radical, where is a natural number. If is even, the expression represents only the positive root.

n n

Example 3: Simplify. a) 4 81 b) 5 32



Entire Radical vs. Mixed Radical

Let’s look examine the radical :

can be expressed as a mixed radical 2 different Methods

Method A (Prime Factorization) Method B (Product of radicals)

Example 4: Express each entire radical as a mixed radical (use method B)

a) b)

c) d)

Example 5:

a)

Express as an entire radical.

b)

Example 6: Without a calculator order the set of numbers from least to greatest.



In all the previous examples so far, the radicand has been a constant term. But what if the radicand contained a variable? Since a variable is representative of a number, we need to ask ourselves if there are any restrictions to what this number can be given the index of the radical.

Let’s take a look at the radical expression What are the values for which this expression is defined?

Let us now look at the expression What are the values for which this expression is defined? Can the expression be simplified?

Example 7: e li

a)

Determine the variable for which the xpressions are defined. Simp fy the expressions.

b) c)

AssignmentPage 278 (min) #1, 2ad, 4, 6bc, 14, (worksheet) 26

(reg) #1 – 4, 6, 14, (worksheet) 26



5.1a Homework Worksheet

26. i. Identify the variables for which the expression is defined

ii. Simplify the expression

a)

b)

c)

Ans: 26a) i. ii.

b) i. ii.

c) i. ii.





5.1b – Adding and Subtracting Radicals

Similar to adding and subtracting polynomials or fractions, radicals have their own set of rules to simplify radicals.

Recall: 1) 3 + 4 = 2) Think : 3)

You are only allowed to add/subtract polynomials that have like terms.

Similarly when you are add/subtract radicals, you can only do so when you have like radicals.Like radicals are radicals that have the same index and same radicand.

Similar to the like terms such as question 2) (where we add the coefficients), we add the numbers that are in front of the radical signs.

Example 1: Simplify.

a) 3 2 3 b) 6 7 2 7 4 7 c) 6 2 4 5

In cases where the index and radicand are not the same, we need to simplify the radicals first in order to see if we have any like radicals to add or subtract.


a) 6 27 4 3 b) 4 24 3 54 2 7 6 28

c) 2 98 10 6 8 4 40 d) 3 316 5 54



Some radical expressions contain variables. Before you attempt to add/subtract the expressions, you must state the values for which the expressions are defined.

Example 3: State the values for which the expression is defined, then simplify.

a) b) c)

Example 4: State the values for which the expression is defined, then simplify.

a) cc 944 b) xx 45320

c) 3 23 2 54216 xx d) 33 803202 xx



Example 5: A square is inscribed in a circle. The area of the circle is 40 m2.

a) What is the exact length of the diagonal of the square?

b) Determine the exact perimeter of the square.

AssignmentPage 278 (min) #5cd, 8cd, 9bd, 10bd, 15

(reg) #5, 8 – 10, 15, 19





5.2 – Multiplying and Dividing Radical Expressions

Multiplying Radical Expressions

The rule of multiplying radicals: a b ab 0a, , 0b

Note: The expression 4 5 has the same meaning as 4 5

When multiplying radicals the index must be the same. You will calculate the product of the values outside the radical sign and the product of the values inside the radical sign.

Example 1: S

a)

implify

b) c) 6423 x

When multiplying radical expressions that contain more than one term, we will use the distributive property or F.O.I.L in order to get rid of the brackets.

Example 2: Expand and simplify

a) 365537 10659528



Example 3: y restrictions Expand and simplify. State an for the variables.

a) b)

Dividing Radical Expressions

The rule for dividing radicals: a

b b

a when ,

You can combine two radicals into one or separate 1 radical into 2 separate ones.


a) 2 30

3 b) 4 24

8 18 c)

3

3 3

3

9

x

x

IMPORTANT : Radicals must NEVER be left in the denominator.

If there is a radical in the denominator, the method to get rid of them is called to “RATIONALIZE THE DENOMINATOR”



Given the expression:1

3If we multiply the numerator and denominator by the radical in the denominator, 3 . The radical will move to the numerator and the answer will remain the same.

1

3

Example 5: Rationalize the denominator then simplify.

a)

b)

c)

Note: Simplify first, it prevents you from doing more work!

Conjugate of a Binomial Expression: The conjugate of is ( )

When the denominator of an expression contains two terms with at least one radical sign, in order to rationalize the denominator, you must multiply both the numerator and denominator by the conjugate of the denominator.



Example 6: Rationalize the denominator. Simplify

a)75

11 b)

AssignmentPage 289, (min) #1be, 2ad, 3bc, 4ad, 5c, 6d, 8c, 10d, 11d, 17 (reg) #1bdef, 2ad, 3bc, 4ade, 5bcd, 6d, 7a, 8c, 10bcd, 11cd, 17





Mid-Unit Activity

1. The area of a rectangle is 16 square units. If the width of the rectangle is units, determine the exact value of the perimeter in simplified radical form with a rationalized denominator.



2. Determine the exact value of both the area and perimeter of the following isosceles triangle.





5.3 – Radical Equations

Radical Equation: __________________________________________________________________

When solving a radical equation we must first state the restrictions on the variable. Because we are taking the square root, there are certain values for the variable that will make the radical expression undefined.

To find the restrictions, knowing that the radicand must be a positive number or zero (radicand ,set up an inequality and solve for the variable

Extraneous Roots: Solution(s) that may arise when an original equation is altered in order to solve the equation. Any “roots” that are not part of the domain of the original equation are called extraneous roots.

To solve a radical equation algebraically:

Step 1: Isolate the radical on one side of the equation. If there are two, isolate the most complex term.

Step 2: Square each side, then solve the equation that results.

(note: If the resulting equation still contains a radical term, repeat steps 1 and 2)

Step 3: Identify extraneous roots and reject them.

Example 1: State the restrictions for , them solve for x .

a) b)



Example 2: State the restrictions for , then solve for x .

a) 3 7 8x 0 b) 3 1x x

Example 3: State the restrictions for , then solve the equation.

3 5x x



Example 4: The period, T , seconds, of a pendulum is related to its length, , in metres. The period is

the time to complete one full cycle and can be approximated with the formula

L

210

LT .

a) Write an equivalent formula with a rational denominator.

b) The length of the pendulum in the HSBC building in downtown Vancouver is 27 m. How long

would the pendulum take to complete 3 cycles (nearest tenth).

Example 5: The mass, , in kilograms, that a beam with a fixed width and length can support is

related to its thickness, , in cetimetres. The formula is

m

t1

5 3

mt , . If a beam is 4 cm thick,

what mass can it support?

0m

Assignment(Min) Page 291, #19ab Page 301, #4bd, 6c, 7d, 8d, 9cd, 10c, 14, 16,18 (Reg) Page 291, #19ab Page 301, #4bd, 5, 6bc, 7cd, 8ad, 9bcd, 10bcd, 14, 16,18





Ch. 5 Review

Key IdeasRestrictions and properties of radical expressions Simplifying radical expressions Adding & subtracting radical expressions Multiplying & Dividing radical expressions Rationalizing the denominator Solving Radical equations

Example 1: Determ e the values of the for which the expres fined. in variables sion is de

a) a) b) c)

Example 2: Simplify the following expressions.

a) b) c)



Example 3: Simplify the fol owing e

a)

l xpressions.

b)

c)



Example 4: R

a)

ationalize the denominator.

b)

c) d)

Example 5: Determine th

a)

e values for which the expressions are defined, then solve.



Example 5: Determine

b)

the values for which the expressions are defined, then solve.

c)

d)

AssignmentPage 304, #1ad, 2cd, 3c, 4ab, 6 (nc), 8, 10bc, 11bc, 13ac, 14 – 17, 18bd, 19ade, 21



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Ch. 5 Notes - WordPress.com€¦ · Pre-Calculus 11 Date: _____ Ch. 5 Radical Expressions & Equations Loo/Lee/Ko Page 1 of 29