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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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 p⁄q.
For any rational number p⁄q, we define: ap⁄q = (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 Math Help tutoring program. There you will find Links to 8 Sets of Problems on Rational Exponents.
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