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 will be equal to . (Except for )
No matter which number we raise to the power of , 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 and in the denominator, the base of the exponent with the positive exponent.
\( 5^0= \)
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:
If we want to simplify the exercise, we get:
Minus times minus = plus
Therefore, the result will be .
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 is even:
When we raise a negative number to an odd power, we get a negative result.
For example:
If we want to simplify the exercise, we get:
Minus times minus = plus
Plus times minus = minus
Therefore, the result will be .
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 is odd:
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:
When the exponent is inside the parentheses, it only applies to the number it belongs to and not to the minus sign before it.
or
orΒ
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:
Click here if you want to learn more about powers of negative numbers.
Any number with an exponent of will be equal to . (Except for )
No matter what number we raise to the power of , we will always get a result of .
Let's see some examples:
Click here to understand the logic behind the rule and learn more about exponents with an exponent of 0.
\( (\frac{1}{4})^{-1} \)
\( 5^{-2} \)
\( 1^0= \)
In an exercise where we have a negative exponent, we turn the term into a fraction where:
The numerator will be and in the denominator, the base of the exponent with the positive exponent.
For example:
We convert the number to a fraction where the numerator is and the denominator is raised to the power of .
We get:
Another example:
We convert the number to a fraction where the numerator is and the denominator is raised to the positive power of .
We get:
Let's move on to a more complex example:
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:
Now, we simply use the ear rule or the division rule between fractions:
We turn it into a multiplication operation and invert the divided fraction. We get:
We solve and get:
We can express as and get:
We use the power of a power rule and get:
Since both bases are identical, we can subtract the exponents according to the quotient rule for exponents with identical bases.
We get:
Note β we could have solved the exercise without the rule and gotten:
Point to ponder:
If we had thought at the beginning of the exercise to turn into , 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:
We will switch the positions of the numerator and the denominator, make the exponent positive, and get:
Click here if you want to learn more about powers with a negative integer exponent.
\( 4^{-1}=\text{?} \)
\( 7^{-24}=\text{?} \)
\( 19^{-2}=\text{?} \)
We use the power property:
We apply it to the problem:
Therefore, the correct answer is C.
We use the power property for a negative exponent:
We will write the fraction in parentheses as a negative power with the help of the previously mentioned power:
We return to the problem, where we obtained:
We continue and use the power property of an exponent raised to another exponent:
And we apply it in the problem:
Therefore, the correct answer is option d.
We use the property of powers of a negative exponent:
We apply it to the problem:
Therefore, the correct answer is option d.
We begin by using the power rule of negative exponents.
We then apply it to the problem:
We can therefore deduce that the correct answer is option B.
Using the rules of negative exponents: how to raise a number to a negative exponent:
We apply it to the problem:
Therefore, the correct answer is option D.