Simplifying Radical Expressions

1

SIMPLIFYING RADICAL EXPRESSIONS

Algebra 1

2

Essential Properties to Understand

Product Property of Like Bases

Division Property of Like Bases

Division Property of Equality

Product Property of Radicals

Quotient Property of Radicals

Power to a Power Property

3


Multiplication of like bases is equal to the base raised to the sum of the exponents.

The base stays the same, even if it is a number.

Keep in mind why we must add exponents here…

4


6 threes being multiplied together, or

Which also comes from

5

Examples

Ex. 1)

Ex. 2)

Ex. 3)

Ex. 4)

Click tab to reveal answer

6

Non-Example A common student mistake is to multiply

the bases when the bases are numbers.

Don’t you make this mistake!

7


Division of like bases is equal to the base raised to the difference of the exponents.

Order of the subtraction matters!

The base stays the same, even if it is a number.

Keep in mind why we must subtract exponents here…

8


Simplify the fraction


9


A power raised to a power is equal to the base raised to the product of the exponents

Keep in mind why we must multiply the exponents here…

10


12 twos being multiplied together, or


11

ExamplesEx. 5)

Ex. 6)

Ex. 7)

* Keep in mind, each member of the group will have the exponent of 3 applied to it using this property.

The 2 does not get the exponent of 4

12

Transition to Radicals Now that we have the basic exponential

properties down, the next 3 slides will introduce properties of radicals.

This will be followed by examples and instruction that apply these new properties.

13


The square root of a product is equal to the product of the square roots of each term.

This property does not apply to expressions that have addition or subtraction under the radicand.

14


The square root of a quotient is equal to the quotient of the square roots of the two terms.

This property will apply even when there are operations applied within the individual radicals being divided, as in

15

Division Property of Equality

Dividing a square root by itself equals 1.

Just as in the Quotient Property of Equality, the operations under the radical can be anything, so long as the numerator and denominator are identical.

16

Simplifying Radicals is Like Simplifying Fractions

In fractions, dividing out the largest factor is most efficient. To be efficient in simplifying radicals, we want to use the largest perfect square value.

17

To Simplify a Radical Expression

Case 1: Take the square root of the perfect square number(s) and/or variable(s)

Case 2: Take the square root of the largest perfect square factors of the

expression, while keeping factors have no square numbers as their factors under the radical.

18

Simple Case 1 Examples

Ex. 8)

Ex. 9)

Ex. 10)

Ex. 11)

Ex. 12)

Goal: To find the largest square value under the radical.

Click tab to reveal answer

19

Special Case in Simplifying If a variable expression with one or more

odd exponents results from a square root, absolute value bars must enclose the expression to ensure no complex numbers result from taking the square root of a negative value.

For example:

20

Simplifying Radical ExpressionsEx. 13)

Factor 32 into a Perfect Square Number and a Non-Perfect Square Number Product Property of Radicals

Square root of 16

Simplified Expression

Note: Perfect square number is on the left and non-perfect square number is on the right under the radical.

21

Alternative Method to Ex. 13 Similar to the example with numeric

fractions, Example 13 can be simplified by using the perfect square number 4 twice.

This can be called the “Scenic Route”, but it will provide the same correct simplified radical expression.

Notice if we simplify with 4 twice, 4 ∙ 4 = 16; therefore, 16 is the largest perfect square number as a factor of 32.

22

Scenic Route

Simplifying using the perfect square number 4

Simplifying using the perfect square number 4 a 2nd time


23

Simplifying Radical Expressions

For this example, we have options. We could …

a) Simplify each radical first, and then multiply. Keep in mind, there may be a need to simplify after multiplying, or

b) Multiply first, and then simplify. This method could produce large numbers that may be difficult to simplify.

Ex. 14)

24

Simplify First…Method a)

Simply first using the perfect square value 4 in the second radical.

Commutative Property Product Property of RadicalsSimply again using the perfect square value 4.Simplified Expression

25

Multiply First…Method b)


Simply using the largestperfect square value 16.

Note: By simplifying with the largest perfect square value, the number of steps are reduced substantially.


26



Simplify Perfect Square Root

Notice, there are no common factors

between 17 and 144

27

Rationalize the Denominator A radical expression is not simplified when a

radical remains in the denominator of a fraction.

To simplify, the Division Property of Equality must be used

This creates a perfect square value under the radical.

When simplified, a value and/or a variable results; therefore, the radical is no longer part of the expression.

28



1

3Simplify Integers


Rationalize the Denominator

29

Important Points

The 2 in the numerator cannot be simplified with the 6 in the denominator, since the 6 is under the radical.

The 6 under the radical cannot be simplified with the 6 in the denominator; however, the coefficient 2 and the 6 in the denominator can be simplified, since they are both integers.

30

Similar to Example 16, we have options…a) We could simplify each radical then

divide and/or simplify, or

b) We could use the Quotient Property of Radicals, divide first, then simplify further if necessary.


31

Dividing First…Method b) Because 256 is divisible by 32, Method b) is

the most efficient method in this example.


Simplify using the perfect square value 4.Simplified Expression

32

Simplify First…Method a)

Simplify using the perfect square value 16 in both numerator and denominator.

Simplified fraction, but not simplified radical expression

1

4

33

Continuation of Method a)


Simplify Integers 2

1

This would certainly be the Scenic Route!


34

Let’s first break each factor down into square partsand non-square parts using the Product Property of Like Bases.


is the square value

Is a square value

is the square value

35

Rewriting the radical expression in a similar fashion to when just numbers are involved,


Take the square root of the perfect square values.

Don’t forget absolute value bars!

36




The 4 and 2 cannot be simplified!

37


Simplify the perfect square value or in the denominator. Don’t forget to include absolute value bars, since the remaining exponent is odd. 2

1

Simplify Integers


38

Ex. 20)


For this example, we have options. We could …

a) Simplify the radical first, and then divide. Keep in mind, there may be a need to simplify after dividing, or

b) Divide, or simplify the fraction first, and then simplify the radical.

39

Simplify the Radical First…Method a)

Quotient Property of Radicals & Rationalize the Denominator

Product Property of Radicals & Multiplication of Like Bases

Separating into Perfect and Non-Perfect Squares in the Numerator, and Simplifyingthe Denominator

40

Continuation of Method a)


Square Root of Perfect Square

4 m p2

3 m p

Simplify Monomial Expressions


41

Simplify the Fraction First…Method b)

Simplify the Monomial Expressions within the Radical


Rationalize theDenominator

Note: You could simplify the radical in the numerator first before rationalizing.

42

Continuation of Method b)

Product Property of Radicals;Separate Numerator into Perfect Square and Non-Perfect Square

Square Root of Perfect Square& Simplified Expression

Simplifying Radical Expressions

Documents

Transcript of Simplifying Radical Expressions