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No credit card required Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
281
Rational Exponents
Fractional powers are radicals in disguise
13.13 Write x
a/b
as a radical expression. Assume that a and b are natural numbers.
Rational exponents (that is, fractional exponents) indicate radical expressions.
The denominator of the rational power (b in this problem) represents the
index of the radical, and the numerator of the power represents either the
exponential power of the radicand or the power to which the entire expression
is raised: and .
13.14 Write 4
3/2
as a radical expression and simplify it.
According to Problem 13.13, . In this problem, x = 4, a = 3, and
b = 2.
13.15 Write 250
1/3
as a radical expression and simplify it.
According to Problem 13.13, . In this problem, x = 250, a = 1, and
b = 3.
Notice that 125 is a perfect cube and a factor of 250.
13.16 Write as an exponential expression with base 7.
The radicand 49 is equal to 7
2
, so . According to Problem 13.13,
. In this problem, x = 7, a = 2, and b = 5.
You can use
either formula
to rewrite x
a/b
.
You’ll probably use
more
often because x
a
tends
to be a large number
that’s hard to
simplify. Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
282
13.17 According to Problem 13.1, . Verify the statement using rational
exponents.
Begin by writing as an exponential expression with base 4.
Rewrite the exponential expression using a rational exponent.
13.18 Write (12x
3
y
2
)
1/2
as a radical expression and simplify.
The entire quantity 12x
3
y
2
is raised to the rational exponent , so construct a
radical expression with radicand 12x
3
y
2
and index 2.
Simplify the radical expression by identifying perfect square factors.
13.19 Write (9x)
1/2
y
7/4
as a radical expression and simplify it.
Note that 9 and x are both raised to the power.
(9x)
1/2
y
7/4
= 9
1/2
x
1/2
y
7/4
Rewrite the expression as a product of radical expressions.
Do this the
same way Problem
13.16 turned
into an exponential
expression with
base 7.

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