Negative Integer Exponents

The exponents are which integer is placed in the power, of base numbers. It can be easily represent as, “a small number to the right side and above of base number”. It is called as exponents. These exponents have some of important rules and laws. Power with negative integer exponents is also one of the rules. Here we are going to explain about this negative integer exponent rule.

If we are having variables, which is containing the exponents and it have equal bases means, we can do some mathematical operations with the exponents. Those operations are called as the “laws of exponents” or “rules of exponents”. In this rule based negative integer rule of exponent is defined as following ways,

Definition for negative integer exponents:

It is otherwise called as power with negative exponent rules. This negative exponent rule is defined as, if m is a positive integer and x is a non-zero rational number, then it can be denoted as,

X-m = (1/x)^m (or)

= (1/x)^m

Which is (x)^-m is the reciprocal of (x)^m

And we adopt the same rule for rational exponents also. If p/q is a positive rational number means, and x>0 is a rational number, then

 X^ - (p/q) = (1/x)^ (p/q) = (1/x)^ (p/q) .

Which is, (x)^-(p/q) is the reciprocal of (x) ^(p/q) or the number obtained by raising the reciprocal of x to the exponent p/q.

For example: 1). 3^-2

= (1/3)^2

= (1/3)^-2

= -6 .

2). (4)^-(2/3)

= (1/4)^-(2/3)

= (1/4)^ (2/3)

This kind of exponentiation used for discovers the negative integer exponents and simplify the problems.


Example problems for negative integer exponents:

1) Solve: (8)^-(2/3)

Solution:

Given: (8)^-(2/3)

= (1/8)^ (2/3)

= [(1/8)^ (1/3)]^ 2

= (1/2)^2, since (1/2)^3 = 1/8

= 1/4 .

2) Solve: (32/243)^-(4/5)

Solution:

Given: (32/243)^-(4/5)

= (243/32)^(4/5)

= [(243/32)^(1/5)]^4

= [(3^5/2^5)^(1/5)]^4

= [((3/2)^5)^(1/5)]^4

= (3/2)^4

= 81/16.

3) Evaluate, and find the following negative integer exponent value:

Evaluate: (27/125)^-(2/3)  xx    (27/125)^-(4/3)

Solution:

 (27/125)^- (2/3) xx (27/125)^-(4/3)

= (125/25)^(2/3) xx (125/27)^(4/3)

= [(5^3/3^3)^(1/3)]^ 2 xx [(5^3/3^3)^(1/3)]^4

= [((5/3)^3)^(1/3)]^2 xx [((5/3)^3)^(1/3)]^4

= (5/3)^2 xx (5/3)^4

= (5/3)^6

= 15625/729.

These all are the explanations and example problems may clear about the negative integer exponents.