We are asked what the Zero Exponent and Negative Exponent Properties mean. We will answer this question by explaining each property one at a time.
Zero Exponent Property
The Zero Exponent Property states that a^0=1, assuming that a ≠0. We can explain this property by looking at the Quotient of Powers Property. Let's consider the division of 2^3 by 2^3. Dividing a number by the same number always results in 1.
2^3 ÷ 2^3 =1
By the Quotient of Powers Property, we know that we can simplify the expression on the left-hand side of the above equation by subtracting the exponents.
2^(3-3) =1 ⇔ 2^0 =1
As we can see, the Zero Exponent Property means that any number other than 0 that is raised to the power of 0 is equal 1.
Negative Exponent Property
The Negative Exponent Property states that a^(- n)= 1a^n, assuming that a ≠0. We can explain this property by looking at the Quotient of Powers Property again. Let's consider the division of 2^2 by 2^4. We know that 2^2=4 and 2^4=16.
2^2 ÷ 2^4 = 2^2/2^4 = 4/16 = 1/4
By the Quotient of Powers Property, we can simplify the expression on the left-hand side by subtracting the exponents.
2^2 ÷ 2^4 = 2^(2-4)= 2^(-2)
As we can see, we get that 2^(-2) is equal to 14, which we can also rewrite as 12^2.
2^(-2) = 1/4 = 1/2^2
The Negative Exponent Property means that a^(- n) is equal to 1a^n.
