Exponents - Special Cases

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 – 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 – 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
$-(5^2 )=$
or 
$-5^2=$
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.

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 – 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 $2^2$ 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 – 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.