Zero and Negative Exponents: Meaning and Evaluation | Grade 8 Math | LevelKite

You already know how to read a power such as $2^4$ . The base is the number being multiplied, here $2$ . The exponent is the small raised number, here $4$ . A positive whole-number exponent counts factors of the base, so $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$ .

So what could $2^0$ or $2^{-3}$ mean? You can't write a product with negative three factors, and it isn't obvious what a product with zero factors should equal. Counting factors stops working, so these powers need a meaning that fits with everything you already know about exponents.

In this lesson you'll find that meaning by following a pattern. Then you'll use it to evaluate powers, which means finding the single number each one equals. Start with the table below. Its first four rows are powers you can already work out.

Powers of 2 as the exponent goes down by 1

Power	Previous value divided by 2	Value
$2^4$	start: $2 \cdot 2 \cdot 2 \cdot 2$	$16$
$2^3$	$16 \div 2$	$8$
$2^2$	$8 \div 2$	$4$
$2^1$	$4 \div 2$	$2$
$2^0$	$2 \div 2$	$1$
$2^{-1}$	$1 \div 2$	$\frac{1}{2}$
$2^{-2}$	$\frac{1}{2} \div 2$	$\frac{1}{4}$
$2^{-3}$	$\frac{1}{4} \div 2$	$\frac{1}{8}$

Look at the first four rows. Each time the exponent goes down by $1$ , you divide the value by the base, $2$ . That makes sense: $2^3$ has one fewer factor of $2$ than $2^4$ .

The last four rows keep dividing by $2$ , and that gives the new powers their values:

$2^0 = 2 \div 2 = 1$ . So $2^0$ is $1$ , not $0$ .
$2^{-1} = \frac{1}{2}$ , $2^{-2} = \frac{1}{4} = \frac{1}{2^2}$ and $2^{-3} = \frac{1}{8} = \frac{1}{2^3}$ .

Each negative power of $2$ is $1$ divided by the matching positive power. That is a reciprocal: the reciprocal of a nonzero number is $1$ divided by that number, so $\frac{1}{8}$ is the reciprocal of $8$ .

Here is one way to picture all three kinds of exponent. Start at $1$ . The exponent $3$ says to multiply by $2$ three times. The exponent $-3$ says to divide by $2$ three times. The exponent $0$ says to do neither, so you stay at $1$ .

A property you already know points to the same meanings. The quotient property says that for a nonzero base $a$ and positive whole-number exponents $m$ and $n$ , $\frac{a^m}{a^n} = a^{m-n}$ . The base can't be $0$ , because then the denominator $a^n$ would be $0$ , and division by $0$ is undefined.

When $m$ is greater than $n$ , the new exponent $m - n$ is positive, and you know how to read it. Suppose you want the property to keep working in the other two cases as well.

Equal exponents. Any nonzero number divided by itself is $1$ , so $\frac{5^3}{5^3} = \frac{125}{125} = 1$ . The quotient property would turn the same fraction into $5^{3-3} = 5^0$ . Both are true only if $5^0 = 1$ .

Larger exponent in the denominator. Write out the factors of $\frac{5^2}{5^4}$ . Then divide the numerator and the denominator by the two factors of $5$ they share:

\frac{5^2}{5^4} = \frac{5 \cdot 5}{5 \cdot 5 \cdot 5 \cdot 5} = \frac{1}{5 \cdot 5} = \frac{1}{5^2}

The quotient property would turn that same fraction into $5^{2-4} = 5^{-2}$ . Both are true only if $5^{-2} = \frac{1}{5^2}$ , which is $\frac{1}{25}$ .

So the values in the table were not a lucky guess. Mathematicians give zero and negative exponents exactly the meanings that keep the quotient property true when the exponents are equal or when the larger exponent is in the denominator. Nothing in this reasoning was special to the bases $2$ and $5$ . It works the same way for every nonzero base, and that leads to the two rules below.

Key concept

Zero and negative exponents

For any nonzero base $a$ and any positive whole number $n$ :

Zero exponent: $a^0 = 1$ .
Negative exponent: $a^{-n} = \frac{1}{a^n}$ , the reciprocal of $a^n$ .

A negative exponent means divide by the base instead of multiplying. It does not make the value negative: a reciprocal keeps the sign of the number, so $a^{-n}$ has the same sign as $a^n$ .

The base cannot be $0$ . Since $0^n = 0$ , the expression $0^{-n}$ would mean $\frac{1}{0}$ , and division by $0$ is undefined. The reasoning for $a^0 = 1$ also divides by the base, so this lesson gives no value to $0^0$ .

a^0 = 1 \quad \text{and} \quad a^{-n} = \frac{1}{a^n} \quad \text{for } a \neq 0

Worked example

A zero exponent and a negative exponent

Evaluate $6^0$ and $4^{-3}$ .

Start with the first power. Its base is 6, which is not 0, so the zero-exponent rule applies: a nonzero base raised to the exponent 0 equals 1.
$6^0 = 1$
Move to the second power. Its base is 4, which is not 0, so the negative-exponent rule applies. Rewrite the power as 1 over the same base with the positive exponent 3.
$4^{-3} = \frac{1}{4^3}$
Evaluate the positive power in the denominator. The exponent 3 counts three factors of 4.
$4^3 = 4 \cdot 4 \cdot 4 = 64$
Replace the denominator with 64. The result is positive, because 1 divided by a positive number is positive. Keep it as a fraction, which is the exact value.
$4^{-3} = \frac{1}{64}$

Answer

$6^0$ equals $1$ , and $4^{-3}$ equals $\frac{1}{64}$ .

Before you evaluate, decide exactly what the base is. A negative sign is part of the base only when parentheses put it there.

In $(-5)^0$ the base is $-5$ . It is not $0$ , so $(-5)^0 = 1$ .
In $-5^0$ there are no parentheses, so the base is $5$ , which is also not $0$ . The negative sign in front means the opposite of $5^0$ , so $-5^0 = -(5^0) = -1$ .

The exponent acts on its base first, and you apply a negative sign outside the power last. The next example reads a negative exponent the same way.

Worked example

Negative signs with and without parentheses

Evaluate $(-2)^{-4}$ and $-2^{-4}$ .

In the first expression the parentheses make negative 2 the base. That base is not 0, so the negative-exponent rule applies: write 1 over the same base with the positive exponent 4.
$(-2)^{-4} = \frac{1}{(-2)^4}$
Evaluate the denominator. Multiply the four factors of negative 2 in pairs. Each pair has the positive product 4, because a negative number times a negative number is positive.
$(-2)^4 = (-2)(-2) \cdot (-2)(-2) = 4 \cdot 4 = 16$
Replace the denominator with 16. That finishes the first expression.
$(-2)^{-4} = \frac{1}{16}$
In the second expression there are no parentheses, so the base is 2 and the negative sign stays outside the power. The base 2 is not 0, so apply the negative-exponent rule to the power first.
$2^{-4} = \frac{1}{2^4} = \frac{1}{2 \cdot 2 \cdot 2 \cdot 2} = \frac{1}{16}$
Now apply the negative sign in front, which means the opposite of the power. The value is negative because of that sign, not because of the negative exponent.
$-2^{-4} = -\frac{1}{16}$

Answer

$(-2)^{-4}$ equals $\frac{1}{16}$ , but $-2^{-4}$ equals $-\frac{1}{16}$ . The parentheses decide whether the negative sign is part of the base.

Worked example

A fraction as the base

Evaluate $\left(\frac{2}{3}\right)^{-3}$ .

The parentheses make the whole fraction two-thirds the base, and it is not 0. Apply the negative-exponent rule: write 1 over the same base with the positive exponent 3.
$\left(\frac{2}{3}\right)^{-3} = \frac{1}{\left(\frac{2}{3}\right)^{3}}$
Evaluate the positive power. Three factors of the fraction means you multiply the three numerators together and the three denominators together.
$\left(\frac{2}{3}\right)^{3} = \frac{2 \cdot 2 \cdot 2}{3 \cdot 3 \cdot 3} = \frac{8}{27}$
Divide 1 by that value. Dividing by a nonzero fraction gives the same result as multiplying by its reciprocal, so the fraction flips.
$1 \div \frac{8}{27} = 1 \cdot \frac{27}{8} = \frac{27}{8}$
Check that the size is reasonable. The base is between 0 and 1, so dividing 1 by it three times should give a number greater than 1. Writing the answer as a mixed number shows that it does.
$\frac{27}{8} = 3\tfrac{3}{8}$

Answer

$\left(\frac{2}{3}\right)^{-3}$ equals $\frac{27}{8}$ , which is the same as $3\tfrac{3}{8}$ .

The takeaway: zero and negative exponents continue a pattern you already knew. For a nonzero base, each step down in the exponent divides the value by the base. That is why, for any nonzero base $a$ and positive whole number $n$ , $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ , the reciprocal of $a^n$ .

To evaluate one of these powers, read the parentheses to find the base, check that the base is not $0$ , rewrite a negative exponent as $1$ over the positive power, and then do the arithmetic. A negative exponent turns multiplying into dividing. It leaves the sign of the value alone: $a^{-n}$ has the same sign as $a^n$ .

Next, you'll use these meanings to simplify expressions with negative exponents and to find and correct errors in applying exponent properties.