Earlier we have seen powers and exponents and their roles. Negative exponents tell us that the power of a number is negative and to make it positive, we need to reciprocal the base. 

 We know that whenever exponents are present, the base is multiplied by the number of times exponents are given. 

 For example, 4^3 (here, the symbol “^” means “raise to” or the power of some number) so 4x4x4=64. In our case, 4 is the base and the power 3 is the exponent. Hence, four raised to 3 which means four is to be multiplied three times. 

 However, in negative exponents, we have to reciprocal the base, initially, and make the exponent positive. Later on, we have to multiply the base based on the number of times the exponent is given. 

 Solve (- 3/ 4) ^-3.

Since (- 3/ 4) ^-3 is given, with -3 as the power, initially, we need to take reciprocal of (-3/4 ).

Then the expression ( -¾ ) ^ (-3) becomes (-4/3) ^ 3.

 The reciprocal of (- 3/ 4) is (-4/3) and then multiply thrice as the exponent or the power is 3.

=(-4/3)^3

= (-4/ 3) x (-4/ 3) x (-4/ 3) 

= (-64/27) 

 Certain rules, based on base and exponent. 

If the exponent is negative then first, we need to reciprocal the base.

If an exponent is an odd number and the base is negative, then the answer will be a negative number. 

If an exponent is an even number and the base is negative then the answer will be a positive number. 

If the base is negative and so is the exponent, the final number will be a positive number.

 Case 1:

The base is negative and the exponent is a negative odd number. 

Solve: (-5/7) ^ (-3)

= (-7/5) ^3 (Initially, we will find reciprocal of the base to make exponent positive) 

= (-7/5) x (-7/5) x (-7/5) 

= (-343/125) 

(Here, the answer is negative as power is odd and base is negative) 

Case 2:

The base is negative and so is the exponent, the answer is a positive number.

= (-11) ^-4 

= (-1/ 11) ^4 (first, we will have to reciprocal the base to make exponent positive) 

= (-1/11) x (-1/11) x (-1/11) x (-1/11) 

= 1/14641

(Here, the answer is positive as power is even and the base is negative) 

 Problem: (3x + 4y) ^-3

Solution:

  If we have (3x + 4y) ^-3, the first step is to reciprocal as to make exponent positive.

= (1 / 3x+4y)^3 .

 Problem: 4^-2 + 2^-4

Solution: 

= (-1 / 4)^2   +  (-1 /2 )^4    {first step is to reciprocal as to make exponent positive}

= (1/16) + (1/16)                 {since denominators are same so no need to find lcm}

= (1+1/16) 

= ( 2/16)                              {after reducing the numerator and denominator}

= (1/ 8) 

 Important points to remember regarding base and exponent: 

Exponents or power means the number of times needs to be multiplied by itself as per the power given. 

b ^-a  = 1/b X 1/b  X  1/b ………. a number of times. 

b ^-n  it is also known as the multiplicative inverse of b^n. 

If b^-c = b^-a then  c = a.

The relationship between the power ( with positive powers) and the negative power is expressed as b^x = (1/b)^-x.

 Difficult Problem: Solve (4^2) x (-2^-4/ 9^-2) 

Solution: First, we need to convert all the negative exponents to positive. 

 As per the law of negative exponent, the expression -2^-4 can be written as ( ½)^4 and (1/9)^-2 can be written as 9^2.

 Hence, we will get the expression as: 

= 4^2 x (9^2 /-2 ^4)

= (4x4)  x  ( 9x9) / (-2 x -2 x -2 x-2) 

Here -2 raised to 4 which is an even number. So,

 the answer while multiplying the

number -2 four times by itself we will get a positive number.

= 16 x 81/ 16    

= 81.