Algebra 1
1
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
2
Product Property of Like Bases

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…
3
Product Property of Like Bases
6 threes being multiplied together, or
Which also comes from
4
Examples
Click tab to reveal answer
Ex. 1)
Ex. 2)
Ex. 3)
Ex. 4)
5
Non-Example

A common student mistake is to multiply
the bases when the bases are numbers.

Don’t you make this mistake!
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Division Property of Like Bases

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…
7
Division Property of Like Bases
Simplify the fraction
Which also comes from
8
Power to a Power Property

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…
9
Power to a Power Property
12 twos being multiplied together, or
Which also comes from
10
Examples
Ex. 5)
The 2 does
not get the
exponent
of 4
Ex. 6)
Ex. 7)
* Keep in mind, each member
of the group
will have
the exponent of 3 applied to it
using this property.
11
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.
12
Product Property of Radicals

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.
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Quotient Property of Radicals

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
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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.
15
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.
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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.
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Simple Case 1 Examples
Click tab to reveal answer
Ex. 8)
Ex. 9)
Ex. 10)
Goal:
To find the largest
square value
under the radical.
Ex. 11)
Ex. 12)
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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:
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Simplifying Radical Expressions
Ex. 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.
20
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.
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Scenic Route
Simplifying using the perfect
square number 4
Simplifying using the perfect
square number 4 a 2nd time
Simplified Expression
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Simplifying Radical Expressions
Ex. 14)
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.
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Simplify First…Method a)
Simply first using the
perfect square value 4 in the
second radical.
Commutative Property
Product Property of Radicals
Simply again using the
perfect square value 4.
Simplified Expression
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Multiply First…Method b)
Product Property of Radicals
Simply using the largest
perfect square value 16.
Simplified Expression
Note: By simplifying with the largest perfect
square value, the number of steps are
reduced substantially.
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Simplifying Radical Expressions
Ex. 15)
Notice, there are no
common factors
between 17 and 144
Quotient Property of Radicals
Simplify Perfect Square Root
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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.
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Simplifying Radical Expressions
Ex. 16)
Rationalize the Denominator
Product Property of Radicals
1
Simplify Integers
3
Simplified Expression
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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.
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Simplifying Radical Expressions
Ex. 17)
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.
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Dividing First…Method b)

Because 256 is divisible by 32, Method b) is
the most efficient method in this example.
Quotient Property of Radicals
Simplify using the perfect
square value 4.
Simplified Expression
31
Simplify First…Method a)
4
1
Simplify using the perfect square
value 16 in both numerator and
denominator.
Simplified fraction, but not
simplified radical expression
32
Continuation of Method a)
Rationalize the Denominator
2
Simplify Integers
1
Simplified Expression
This would certainly
be the Scenic Route!
33
Simplifying Radical Expressions
Ex. 18)
Let’s first break each factor down into square parts
and non-square parts using the Product Property
of Like Bases.
is the square value



Is a square value
is the square value
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Simplifying Radical Expressions
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!
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Simplifying Radical Expressions
Ex. 19)
The 4 and 2 cannot
be simplified!
Rationalize the Denominator
Product Property of Radicals
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Simplifying Radical Expressions
Simplify the perfect square value
or
in the denominator.
Don’t forget to include absolute
value bars, since the remaining
exponent is odd.
2
Simplify Integers
1
Simplified Expression
37
Simplifying Radical Expressions
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.
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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 Simplifying
the Denominator
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Continuation of Method a)
Product Property of Radicals
Square Root of Perfect Square
4 m p2
Simplify Monomial Expressions
3mp
Simplified Expression
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Simplify the Fraction First…Method b)
Simplify the Monomial
Expressions within
the Radical
Quotient Property
of Radicals
Rationalize the
Denominator
Note: You could simplify the radical in the
numerator first before rationalizing.
41
Continuation of Method b)
Product Property of Radicals;
Separate Numerator into
Perfect Square and NonPerfect Square
Square Root of Perfect Square
& Simplified Expression
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