Exponents - Special Cases

πŸ†Practice powers - special cases

Powers - Special Cases

Powers of negative numbers

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.

Powers with exponent \(0\)

Any number with an exponent of 00 will be equal to 11. (Except for 00)
No matter which number we raise to the power of 00, we will always get a result of 1.

Powers with negative integer exponents

In an exercise where we have a negative exponent, we turn the term into a fraction where:
the numerator will be 11 and in the denominator, the base of the exponent with the positive exponent.

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\( (-2)^7= \)

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Exponents - Special Cases

Powers of negative numbers

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.

Raising a negative number to an even power

When we raise a negative number to an even power, we get a positive result.
For example:
(βˆ’4)2=(-4)^2=
If we want to simplify the exercise, we get:
(βˆ’4)βˆ—(βˆ’4)=(-4)*(-4)=
Minus times minus = plus
Therefore, the result will be 1616.
Essentially – if the number is negative and the power is even, we can ignore the minus.
Let's formulate this as a rule:
When nn is even:
(βˆ’x)n=xn(-x)^n=x^n

Raising a negative number to an odd power

When we raise a negative number to an odd power, we get a negative result.
For example:
(βˆ’4)3=(-4)^3=
If we want to simplify the exercise, we get:
(βˆ’4)βˆ—(βˆ’4)βˆ—(βˆ’4)=64(-4)*(-4)*(-4)=64

Minus times minus = plus
Plus times minus = minus
Therefore, the result will be 64βˆ’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 nn is odd:
(βˆ’x)n=βˆ’(x)n(-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:
(βˆ’5)2=(-5)^2=
(βˆ’5)βˆ—(βˆ’5)=25(-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.
(βˆ’52)=(-5^2 )=
or
βˆ’(52)=-(5^2 )=
orΒ 
βˆ’52=-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:
βˆ’52=βˆ’25-5^2=-25

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

Powers to the zero power

Any number with an exponent of 00 will be equal to 11. (Except for 00)
No matter what number we raise to the power of 00, we will always get a result of 11.
Let's see some examples:
50=15^0=1
5.8970=15.897^0=1
100000=110000^0=1
(23)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.

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Powers with a negative integer exponent

In an exercise where we have a negative exponent, we turn the term into a fraction where:
The numerator will be 11 and in the denominator, the base of the exponent with the positive exponent.

For example:
3βˆ’2=3^{-2}=
We convert the number to a fraction where the numerator is 11 and the denominator is 33 raised to the power of 22.
We get:
132\frac{1}{3^2}

Another example:
6βˆ’3=6^{-3}=
We convert the number to a fraction where the numerator is 11 and the denominator is 66 raised to the positive power of 33.
We get:
163\frac{1}{6^3}

Let's move on to a more complex example:
2βˆ’34βˆ’2=\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:
123142\frac{1}{2^3}\over\frac{1}{4^2}
Now, we simply use the ear rule or the division rule between fractions:
123:142=\frac{1}{2^3}:\frac{1}{4^2}=

We turn it into a multiplication operation and invert the divided fraction. We get:
123β‹…421=\frac{1}{2^3}\cdot\frac{4^2}{1}=
We solve and get:
4223=\frac{4^2}{2^3}=
We can express 44 as 22Β 2^2 Β  and get:
(22)223=\frac{(2^2)^2}{2^3}=
We use the power of a power rule and get:
24232^4\over2^3
Since both bases are identical, we can subtract the exponents according to the quotient rule for exponents with identical bases.
We get:
21=22^1=2
Note – we could have solved the exercise without the rule and gotten:
2423=168=2\frac{2^4}{2^3}=\frac{16}{8}=2

Point to ponder:
If we had thought at the beginning of the exercise to turn 424^2 into 242^4, 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.

What do you do when you have a fraction with a negative exponent?

We will switch the positions of the numerator and the denominator and make the exponent positive.
For example:
(68)βˆ’2=(\frac{6}{8})^{-2}=
We will switch the positions of the numerator and the denominator, make the exponent positive, and get:
(86)2=(\frac{8}{6})^{2}=

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

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