Radicals
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Radicals
We know what 2n means whenever n is an integer. To give
meaning to a power, such as 24/5, whose exponent is a
rational number, we need to discuss radicals.
The symbol
means “the positive square root of.” Thus
= b means b2 = a and b  0
Since a = b2  0, the symbol
a  0. For instance,
=3
because
makes sense only when
32 = 9 and
30
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Radicals
Square roots are special cases of nth roots. The nth root of
x is the number that, when raised to the nth power, gives x.
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Radicals
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Example 8 – Simplifying Expressions Involving nth Roots
(a)
Factor out the largest cube
Property 1:
Property 4:
(b)
Property 1:
Property 5,
Property 5:
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Example 9 – Combining Radicals
(a)
Factor out the largest squares
Property 1:
Distributive property
(b) If b > 0, then
Property 1:
Property 5, b > 0
Distributive property
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Rational Exponents
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Rational Exponents
To define what is meant by a rational exponent or,
equivalently, a fractional exponent such as a1/3, we need to
use radicals. To give meaning to the symbol a1/n in a way
that is consistent with the Laws of Exponents, we would
have to have
(a1/n)n = a(1/n)n = a1 = a
So by the definition of nth root,
a1/n =
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Rational Exponents
In general, we define rational exponents as follows.
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Example 11 – Using the Laws of Exponents with Rational Exponents
(a) a1/3a7/3 = a8/3
(b)
Law 1: ambn = am +n
= a2/5 + 7/5 – 3/5
Law 1, Law 2:
= a6/5
(c) (2a3b4)3/2 = 23/2(a3)3/2(b4)3/2
=(
=2
)3a3(3/2)b4(3/2)
Law 4: (abc)n = anbncn
Law 3: (am)n = amn
a9/2b6
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Example 11 – Using the Laws of Exponents with Rational Exponents
(d)
Laws 5, 4, and 7
Law 3
Law 1, Law 2
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Rationalizing the Denominator
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Rationalizing the Denominator
It is often useful to eliminate the radical in a denominator by
multiplying both numerator and denominator by an
appropriate expression. This procedure is called
rationalizing the denominator.
If the denominator is of the form
, we multiply numerator
and denominator by
. In doing this we multiply the given
quantity by 1, so we do not change its value. For instance,
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Rationalizing the Denominator
Note that the denominator in the last fraction contains no
radical. In general, if the denominator is of the form
with m < n, then multiplying the numerator and denominator
by will rationalize the denominator, because
(for a > 0)
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Example 13 – Rationalizing Denominators
(a)
(b)
(c)
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