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.

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 = 3^1⁄2
and is called 3 power one by two or 2nd root or square root of 3.
3^1⁄2 is also denoted by √3.
In other words,
3^1⁄2 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,
a^1⁄2 called a power one by two or 2nd root or square root of a
is the real number x such that x² = a.
Similarly
a^1⁄3 called a power one by three or 3rd root or cube root of a
is the real number x such that x³ = a.
a^1⁄4 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,

a^1⁄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 4^1⁄2 is always taken as +2.

In general,

for positive value of a,
The value of a^1⁄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 a^1⁄n does n't have a real value.

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

For any rational number p⁄q, we define:
a^p⁄q = (a^1⁄q)^p = (a^p)^1⁄q = qth root of a^p.

The 7 Laws of Exponents and the 2 Rules of Exponents
given in Laws of Exponents are valid for Rational Exponents also.

Sets of Solved Examples and Exercises: Rational Exponents

For Solved Examples and Exercise
Problems on Application of the Laws
to Rational Exponents, Go to

Set1 of Solved Examples and Exercise Problems

Set2 of Solved Examples and Exercise Problems

Set3 of Solved Examples and Exercise Problems

Set4 of Solved Examples and Exercise Problems

Set5 of Solved Examples and Exercise Problems

Set6 of Solved Examples and Exercise Problems

Set7 of Solved Examples and Exercise Problems

Set8 of Solved Examples and Exercise Problems