What is negative Exponents?
The Negative Exponent Rule in the laws of exponents states that any base with a negative exponent can be expressed as the reciprocal of the base with a positive exponent. In other words, for any non-zero number a and exponent n the rule is:
Rule 1: a^−n=1/a^n
If the base is already in the denominator, a negative exponent can move it to the numerator:
Rule 2: 1/a^-n=a^n
For this rule you are make positive the exponent by making the reciprocal:
Rule 3: (a/b)^-n= (b/a)^n
Examples of Rule 1:
x^-2=1/x^2
For this example, we did the reciprocal of x^-2 which is 1/x^2
4^-3= 1/4^3=1/64
In this example, we can see that we add an coefficient and not a variable. So, the first thing that we did was the reciprocal which gave us 1/4^3 and then simplify the 4 as 4*4*4 which is 64.
(5a)^-2= 1/(5a)^2=1/25a^a
So for this examples, we put coefficient and variables. First the reciprocal which is 1/(5a)^2
Then we Simplify which is 1/(5*5)* 1/(a^2) so the answer will be 1/25a^2.
Example of Rule 2:
1/a^-4=a^4
So, fort his example of rule 2 we first find the reciprocal as well. Whenever we find the reciprocal, we can start and simplify. Which gave us a^4.
1/4^-3= 4^3=64
In this example we first find the reciprocal which was 4^3 and gave us 64, or just leave it as 4^3. Both answers are correct.
Example for Rule 3:
(4/6)^-2= (6/4)^2=6^2/4^2=36/16=9/4
For this Example we can see that we have coefficient and not variables. But it is the same thing, first we find the reciprocal which is (6/4)^2 and then apply the exponent to the bases which is 6^2/4^2 and then we simplify which is 36/16 and lastly, we divide which the answer is 9/4.
(3a/5b)^-3=(5b/3a)^3=(5b)^3/(3a)^3=125b^3/27a^3
So, for this example we are using coefficient and variables together. So, first we find what is the reciprocal which is (5b/3a)^3. Then we apply the exponent to the bases which is (5b)^3/(3a)^3 and lastly, we simplify (5*5*5) * (b^1*3)/(3*3*3)*(a^1*3) which gives us 125b^3/27a^3.
Reflection
Reciprocals are represented by negative exponents. When dealing with expressions that contain inverse terms, it is crucial to comprehend this rule. It is also a useful idea in math courses when rearranging equations or transferring terms between fractions.