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.
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.
Research-based personalized Math Help tutoring program : Rational Exponents
Here is a resource for Solid Foundation in Math Fundamentals from Middle thru High School. You can check your self by the
There is an exclusive, Parent Information Page provides YOU with detailed reports of your child’s progress so you can monitor your child’s success and give them encouragement. These Reports include
Time spent using the program
Personalized remediation curriculum designed for your child
Details the areas of weakness where your child needs additional help
Provides the REASONS WHY your child missed a concept
List of modules accessed and amount of time spent in each module
Creates reports that can be printed and used to discuss issues with your child’s teachers
These reports are created and stored in a secure section of the program, available exclusively to you, the parent. The section is accessed by a password that YOU create and use. No unauthorized users can access this information.
Recently, I have found a series of math curricula (Both Hard Copy and Digital Copy) developed by a Lady Teacher who taught everyone from Pre-K students to doctoral students and who is a Ph.D. in Mathematics Education.
This series is very different and advantageous over many of the traditional books available. These give students tools that other books do not. Other books just give practice. These teach students “tricks” and new ways to think.
These build a student’s new knowledge of concepts from their existing knowledge. These provide many pages of practice that gradually increases in difficulty and provide constant review.
These also provide teachers and parents with lessons on how to work with the child on the concepts.
The series is low to reasonably priced and include