When we have an exponent on a negative number, we can get a positive result or a negative result.

We will know this based on the exponent β whether it is even or odd.

When we have an exponent on a negative number, we can get a positive result or a negative result.

We will know this based on the exponent β whether it is even or odd.

Any number with an exponent of $0$ will be equal to $1$. (Except for $0$)

No matter which number we raise to the power of $0$, we will always get a result of 1.

In an exercise where we have a negative exponent, we turn the term into a fraction where:

the numerator will be $1$ and in the denominator, the base of the exponent with the positive exponent.

\( (-2)^7= \)

When we have an exponent on a negative number, we can get a positive result or a negative result.

We will know this based on the exponent β whether it is even or odd.

When we raise a negative number to an even power, we get a positive result.

For example:

$(-4)^2=$

If we want to simplify the exercise, we get:

$(-4)*(-4)=$

Minus times minus = plus

Therefore, the result will be $16$.

Essentially β if the number is negative and the power is even, we can ignore the minus.**Let's formulate this as a rule:**

When $n$ is even:

$(-x)^n=x^n$

When we raise a negative number to an odd power, we get a negative result.

For example:

$(-4)^3=$

If we want to simplify the exercise, we get:

$(-4)*(-4)*(-4)=64$

Minus times minus = plus

Plus times minus = minus

Therefore, the result will be $64-$.

Essentially, if the number is negative and the exponent is odd, we cannot ignore the minus and will always get a negative result.** Let's formulate this as a rule:**

When $n$ is odd:

$(-x)^n=-(x)^n$

**Note!** There is a huge difference if the exponent is inside the parentheses versus if the exponent is outside the parentheses!

When the exponent is outside the parentheses - it acts on everything inside the parentheses.

Like in the following exercise:

$(-5)^2=$

$(-5)*(-5)=25$

When the exponent is inside the parentheses, it only applies to the number it belongs to and not to the minus sign before it.

$(-5^2 )=$

or

$-(5^2 )=$

orΒ

$-5^2=$

The exponent refers only to the number and not to the minus sign before it.

Therefore, we calculate the exponent and add the minus as a kind of addition.

We get:

$-5^2=-25$

Click here if you want to learn more about powers of negative numbers.

Any number with an exponent of $0$ will be equal to $1$. (Except for $0$)

No matter what number we raise to the power of $0$, we will always get a result of $1$.

Let's see some examples:

$5^0=1$

$5.897^0=1$

$10000^0=1$

$(\frac{2}{3})^0=1$

Click here to understand the logic behind the rule and learn more about exponents with an exponent of 0.

Test your knowledge

Question 1

\( \)\( -(2)^2= \)

Question 2

\( 9= \)

Question 3

\( \)\( (-8)^2= \)

In an exercise where we have a negative exponent, we turn the term into a fraction where:

The numerator will be $1$ and in the denominator, the base of the exponent with the positive exponent.

**For example:**

$3^{-2}=$

We convert the number to a fraction where the numerator is $1$ and the denominator is $3$ raised to the power of $2$.

We get:

$\frac{1}{3^2}$

**Another example:**

$6^{-3}=$

We convert the number to a fraction where the numerator is $1$ and the denominator is $6$ raised to the positive power of $3$.

We get:

$\frac{1}{6^3}$

**Let's move on to a more complex example:**

$\frac{2^{-3}}{4^{-2}}=$

We know that the exercise looks a bit intimidating, but if we follow the rule we learned, we can solve it quite easily.**Remember that the rules do not change** β when there is a base with a negative exponent, it turns into a fraction according to the rules we learned. We will turn each term into a fraction and get:

$\frac{1}{2^3}\over\frac{1}{4^2}$

Now, we simply use the ear rule or the division rule between fractions:

$\frac{1}{2^3}:\frac{1}{4^2}=$

We turn it into a multiplication operation and invert the divided fraction. We get:

$\frac{1}{2^3}\cdot\frac{4^2}{1}=$

We solve and get:

$\frac{4^2}{2^3}=$

We can express $4$ as $2^2 Β$ and get:

$\frac{(2^2)^2}{2^3}=$

We use the power of a power rule and get:

$2^4\over2^3$

Since both bases are identical, we can subtract the exponents according to the quotient rule for exponents with identical bases.

We get:

$2^1=2$**Note** β we could have solved the exercise without the rule and gotten:

$\frac{2^4}{2^3}=\frac{16}{8}=2$

**Point to ponder:**

If we had thought at the beginning of the exercise to turn $4^2$ into $2^4$, we would have gotten a much easier exercise to solve.

Thus, we would initially create a fraction with identical bases and therefore we could subtract the exponents.

We will switch the positions of the numerator and the denominator and make the exponent positive.

For example:

$(\frac{6}{8})^{-2}=$

We will switch the positions of the numerator and the denominator, make the exponent positive, and get:

$(\frac{8}{6})^{2}=$

Click here if you want to learn more about powers with a negative integer exponent.

Do you know what the answer is?

Question 1

\( 112^0=\text{?} \)

Question 2

\( 4^0=\text{?} \)

Question 3

\( (\frac{1}{8})^0=\text{?} \)

Related Subjects

- Square root of a product
- Square Roots
- Square root of a quotient
- Power of a Quotient
- Exponent of a Multiplication
- Multiplying Exponents with the Same Base
- Division of Exponents with the Same Base
- Power of a Power
- Exponents - Special Cases
- Negative Exponents
- Zero Exponent Rule
- The quadratic function
- Parabola