2. Vocabulary
Radical Expression – expression that contains a
radical, such as a square root, cube root, or other root.
Simplest form of a radical expression:
No perfect square factors are in the radicand.
No fractions are in the radicand.
No radicands appear in the denominator of a fraction.
3. Product Property of Radicals
The square root of a product equals the product of the
square roots of the factors.
ab = a × b where a ³ 0 and b ³ 0
4x = 4 × x = 2 x
4. Example 1 Use the product property of radicals
=
a. 32 16 • 2 Factor using perfect square factor.
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16 • 2 Product property of radicals
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4 2 Simplify.
9 • x2 • x
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b. 9x3 Factor using perfect square factors.
9 • x2 •
=
x Product property of radicals
=
3x x Simplify.
5. Example 2 Multiply radicals
2• 8 2•8
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a. Product property of radicals
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16 Multiply.
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4 Simplify.
3x • 4 x 4 3x • x
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b. Product property of radicals
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4 3x2 Multiply.
4• 3•
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x2 Product property of radicals
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4x 3 Simplify.
6. Example 2 Multiply radicals
7xy2 • 3 x 3 7xy2 • x
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c. Product property of radicals
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3 7x2y2 Multiply.
3• 7• x2 • y2 Product property of radicals
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=
3xy 7 Simplify.
7. Quotient Property of Radicals
The square root of a quotient equals the quotient of
the square roots of the numerator and denominator.
a a
= where a ³ 0 and b > 0
b b
16 16 4
= =
25 25 5
8. Example 3 Use the quotient property of radicals
a. 13 13 Quotient property of radicals
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100 100
13
= Simplify.
10
b. 7 7
2
= Quotient property of radicals
x x2
7 Simplify.
=
x
9. Rationalizing the denominator
The process of eliminating a radical from an
expression’s denominator is called rationalizing the
denominator.
10. Example 4 Rationalize the denominator
5 5 7 7
a. = • Multiply by .
7 7 7 7
5 7
= Product property of radicals
49
5 7
= Simplify.
7
2 2 3b 3b
b. = • Multiply by .
3b 3b 3b 3b
11. Example 4 Rationalize the denominator
6b
= Product property of radicals
9b2
6b
= Product property of radicals
9 • b2
6b
= Simplify.
3b