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 \neq = 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} \div 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 \Leftrightarrow 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} = \frac{1}{a ^{n}},

assuming that

a \neq = 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} \div 2^{4} = \frac{2 ^{2}}{2 ^{4}} = \frac{4}{1 6} = \frac{1}{4}

By the Quotient of Powers Property, we can simplify the expression on the left-hand side by subtracting the exponents.

2^{2} \div 2^{4} = 2^{2 - 4} = 2^{- 2}

As we can see, we get that

2^{- 2}

is equal to

\frac{1}{4},

which we can also rewrite as

\frac{1}{2 ^{2}} .

2^{- 2} = \frac{1}{4} = \frac{1}{2 ^{2}}

The Negative Exponent Property means that

a^{- n}

is equal to

\frac{1}{a ^{n}} .

Exercise