Math 152 — Rodriguez
Blitzer — 7.3
Multiplying and Simplifying Radical Expressions
I. Perfect nth powers
Perfect squares: 1, 4, 9, 16, 25, 36, etc…
x2, x4, x6, x8, x10, …
Perfect cubes:
variable: exponent is
1, 8, 27, 64, 125, etc…
x3, x6, x9, x12, x15, …
Perfect 4th powers:
variable: exponent is
1, 16, 81, 256, 625, etc…
x4, x8, x12, x16, x20, …
Perfect 5th powers:
variable: exponent is
1, 32, 243, etc…
x5, x10, x15, x20, x25, …
variable: exponent is
II. Simplifying Radicals
To simplify a radical, for now, means to remove any nth powers from the radicand. We will
add more rules in the next section.
Product Rule for Radicals states: If
n
a and
n
b are real numbers, then
Use the Product Rule to rewrite the radical as:
n largest
n
a ⋅ n b = n ab .
perf nth power ⋅ n no perf nth power
Examples: Simplify by factoring.
1)
12
5)
2)
48
6)
3)
x5
4)
3
x8
Blitzer — 7.3
40 x 3
3
5 12
−32 x y
7)
4
32 x y
8)
5
−64 x y
7 10
5
12
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9)
3
10) 4 162x 5 y 10
54x 4 y 7
III. Multiplying Radicals
Use the Product Rule to first rewrite the product as one radical expression. Then use
the rule again, to simplify.
Examples: Multiply and simplify. Assume that all variables in a radicand represent
positive real numbers and no radicands involve negative quantities raised to even
powers.
1)
4
12 x y z ⋅ 4 8 x y z
3)
3
4 x y z ⋅ 3 4 x yz
5)
Blitzer — 7.3
4
5
2
3 6
4
6
5
2 6
2)
8
4)
3
10x 4 y 2 z 4 ⋅ 3 8x 6 y 2 z 2
5
3
7
2
3
3 7
8x y z ⋅ 5 8x y z
8x 3y 7 z 2 ⋅ 4 8x 3y 3z 7
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