# RATIONAL EXPONENTS - EXPLANATION, SOLVED EXAMPLES AND EXERCISES

Please study  Laws of Exponents before Rational Exponents,
if you have not already done so.

It is a prerequisite here.

There, we stated the 7 laws of indices
and the two Rules used for solving problems.

We also gave the explanations and proofs
of the 7 laws for whole number exponents.

We provided Solved Examples and Problems for Practice
with Answers to help to apply the 7 Laws and the 2 Rules
in solving problems for whole number exponents.

Here, we provide the explanation for Fractional Exponents.

We apply the same 7 Laws and the 2 Rules in
solving problems for fractional exponents.

We provide Links to a number of solved examples
and problems for practice with answers on Rational Exponents.

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## Explanation for Rational Exponents

Look at the question:
if x2 = 4, what is x ?
You may answer the question as:
Since 4 = 2 x 2 = 22, x2 = 22
and as the bases are equal, the exponents are equal. ∴ x = 2.
Here 4 is a perfect square and you could answer the question easily.

If I give you 3 on the right side instead of 4, i.e. if x2 = 3, what is x ?
How do you answer the question?
The same question can be asked as: Find a number whose square is 3.
The answer lies in the definition of a number with rational exponent.
The solution for x2 = 3 is x = 31⁄2
and is called 3 power one by two or 2nd root or square root of 3.
31⁄2 is also denoted by √3.
In other words,
31⁄2 called 3 power one by two or 2nd root or square root of 3
is the real number x such that x2 = 3.

If we take a (any real number) in place of 3,
a1⁄2 called a power one by two or 2nd root or square root of a
is the real number x such that x2 = a.
Similarly
a1⁄3 called a power one by three or 3rd root or cube root of a
is the real number x such that x3 = a.
a1⁄4 called a power one by four or 4th root of a
is the real number x such that x4 = a.
......................................... and so on.
In general, for any positive ineger n,

a1⁄n called a power one by n or nth root of a
is the real number x such that xn = a.

When I asked you to find x, given x2 = 4, (the question given above),
did you observe that (-2)2 is also 4.
x2 = 4 has two solutions, x = +2 and -2.
You are right.
However 41⁄2 is always taken as +2.

In general,

for positive value of a,
The value of a1⁄n is always taken as positive.

Now I ask you another question:
Can you find a number whose square is -4 ?
We know square of -2 is +4 and not -4.
And there is no real number which
when squared gives negative real number.
That means (-4)1⁄2 does n't have a real value.
Similarly, (-16)1⁄4 does n't have a real value.
In general,

If a is a negative real number and n is an even positive integer,
then a1⁄n does n't have a real value.

Thus (-3)1⁄2, (-49)1⁄2, (-81)1⁄4, (-5)1⁄4, (-64)1⁄6 don't have a real value.

So far we have seen powers like 1⁄n. What about powers like pq.

For any rational number pq, we define:
apq = (a1⁄q)p = (ap)1⁄q = qth root of ap.
Now Let us go to study Rational Exponents.
The 7 Laws of Exponents and the 2 Rules of Exponents
given in Laws of Exponents are valid for Rational Exponents also.
We need those Laws and Rules to
solve problems on Rational Exponents.

## Sets of Solved Examples and Exercises: Rational Exponents

For Solved Examples and Exercise
Problems on Application of the Laws
to Rational Exponents, Go to
the Links given below the Review of
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There you will find Links to 8 Sets of
Problems on Rational Exponents.

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