6.2 Simplifying Radical Expressions

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6.2 Simplifying Radical Expressions
Product rule for radicals is useful in
simplifying radicals.
The product rule can be written as
For example 36 9 4
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Any number that can be written in the form a2 is
a perfect square.
4 22 is a perfect square.
9 32 is a perfect square.
121 112 is a perfect square.
196 142 is a perfect square.
Write some perfect square numbers on your own.
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Any number that can be written in the form a3 is
a perfect cube.
8 23 is a perfect cube.
27 33 is a perfect cube.
64 43 is a perfect cube.
125 53 is a perfect cube.
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Any number of the form a4 is a perfect 4th power.
16 24 is a perfect 4th power.
81 34 is a perfect 4th power.
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It is easy to obtain the square root of a perfect
square.
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To find the square root of a number with an
exponent, divide the exponent by 2
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Similarly it is easy to obtain the cube root of a
perfect cube.
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To find the cube root of a number with an
exponent, divide the exponent by 3.
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We will use the following steps to simplify a
radical expression.

1. Write each factor inside the radical as the
   product of the largest perfect power for the
   radical.

27 is a perfect cube.
16 and x4 are perfect squares
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2. Use the product rule to write the expression
as a product of radicals.

1. Simplify the radical containing perfect powers.

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We write 24 as a product containing the largest
perfect square.
4 is the largest perfect square in 24
24 4 6
Write this as the product of two radicals
Simplify the perfect square radical
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We write 18 as a product containing the largest
perfect square.
9 is the largest perfect square in 18
18 9 2
Write this as the product of two radicals
Simplify the perfect square radical
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We write 24 as a product containing the largest
perfect cube.
8 is the largest perfect cube in 24
24 8 3
Write this as the product of two radicals
Simplify the perfect cube radical
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We write 81 as a product containing the largest
perfect cube.
27 is the largest perfect cube in 81
81 27 3
Write this as the product of two radicals
Simplify the perfect cube radical
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We write b23 as a product containing the largest
perfect 4th power. A perfect 4th power is
divisible by 4
b20 is the largest perfect 4th power in b23
b23 b20 b3
Write this as the product of two radicals
Simplify the perfect 4th power radical
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Write each number inside the radical as the
product of the highest perfect square.
a7 a6 a
b11 b10 b
75 25 3
25, a6 and b10 are perfect squares.
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Write each number inside the radical as the
product of the highest perfect cube.
16 8 3
8 is a perfect cube
x3 and y6 are perfect cubes.
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24 4 6
Here 4 22 is a perfect square
y20 y20
z27 z26 z
x15 x14 x
x14 , y20 and z26 perfect squares as the
exponents can be divided by 2
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This can be simplified as
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This can be written as
21
This can be written under one cube root as
2