We will see several examples of negative exponents
Example 1
4โ3=
We can see that the exponent is a negative number. Therefore, we will convert the expression to a fraction in this way: the numerator will be and in the denominator we will place the base of the power with positive exponent.
1
That is to say:
431โ
If we have a fraction with a 1 in the numerator and a base with some expression in the denominator, we must add a subtraction sign to the exponent in the denominator to convert it into a base with a negative exponent. Only in this way we will obtain the correct exponent.
Example 2
42โX1โ=
In order to write the expression as a power with negative exponent we must add a negative sign outside the exponent in the denominator as follows:
4โ(2โX)=
Then we will take out the parentheses and we will obtain:
4โ2+X=
Another way to do this is to modify the sign of each term.
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Example 3
3โ
5Xโ7=
In the expression we have a negative exponent. We will begin by separating the X from its coefficient 5.
We have:
3โ
5โ
Xโ7=
Now it is clearer to us that to proceed we will have to convert the X into fraction in the way we learned. We will do it and we will obtain:
3โ
5โ
X71โ=
Great. We will multiply the terms and we will get:
X715โ
Example 4
4โ52โ4โ
I'm sure you're saying, "Hey, how am I going to solve this exercise?
That's exactly what we're here for. We will do it slowly and safely.
Remember that the properties do not change: when there is a base with negative exponent it becomes a fraction just as we have learned. We will convert each term to a fraction and get:
451โ241โโ=
Then, simply use the property of division of fractions:
241โ:451โ
We will convert it to a multiplication operation and invert the fraction by which we divide. We will obtain:
241โโ
145โ=
We will solve and get:
2445โ=
We can express the 4 as 22, then we apply the quotient law of exponents with the same base and we will have:
24210โ=
Since we have the same base we can subtract the exponents and we will get:
26=64
Attention!
We could have solved the exercise in a much simpler way if we had thought before to convert the 4 to 22.
If so, we would have created from the beginning a fraction with equal bases and we would have subtracted the exponents.
Shall we try?
Let us recall the original exercise:
4โ52โ4โ=
Now let's put the 4 as 22 We will have:
(22)โ52โ4โ=
Now let's apply to the denominator the power property of a power and arrive at:
(2)โ102โ4โ=
Good. Now we can subtract the exponents since we have equality of bases.
Let's remember that when we multiply negative numbers the result is positive.
We will get:
26=64
And, as you see, we have obtained the same result by a shorter path.
What happens when there is a fraction that is raised totally to a negative exponent?
It is very simple, we will invert the numerator with the denominator and transform the exponent to positive.
Do you know what the answer is?
Let's see an example
(42โ)โ3=
We see that we have a negative exponent that applies to the whole fraction.
Therefore, we will invert the numerator with the denominator, transform the exponent to positive and obtain:
(24โ)3=
Tip: before rushing to apply the exponent to each term it is convenient to look at what expression is between the parentheses. Clearly we have a 2.
Therefore, it will give us:
23=8
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Exercises for negative exponents
Exercise 1
Solve the following exercise:
(41โ)โ1
Solution:
When we have a negative power, we convert the numerator and denominator, thus 41โ converting14โ, i.e., the answer is.4
Answer:
4
Exercise 2
Solve the following exercise:
5โ2
Solution:
According to the property, when we have a whole number with a power that is negative, the number will become a fraction of the original number, when the power will affect the denominator (the one that was the original number).
In other words 5โ2=521โ
Now all that remains is to solve for the power in the denominator.
Answer:
251โ
Exercise 3
Solve the following exercise:
[(71โ)โ1]4=
Solution:
In this exercise there are two parts, and we start by solving according to the order of operations, from the inner parentheses outward.
According to the rule, when it is in a negative power, it is made into a positive by replacing the numerator and denominator.
7โ11โ=71
Now, we use the power law to multiply the power that is enclosed in parentheses with the one inside.
1ร4=4
Thus we arrive at the solution
Answer:
74
Do you think you will be able to solve it?
Exercise 4
Solve the following exercise:
X6X7โ1โ=
Solution:
First we note the fraction in the denominator of the exercise.
Also here two laws are used, first of all the power law, according to which we do
x6x7โ=x(7โ6)=x1=x
Now, we subtract only the fraction
x1โ
We know that this form can also be converted through the property of the negative exponents, so we can also write:
Answer:
x1โ=xโ1
Exercise 5
Simplify the expression ((2y)โ3x2โ)โ2
Solution
First we solve the expression inside the parentheses. The denominator has a negative exponent so we can place it in the numerator and change the sign of the exponent.
((2y)โ3x2โ)โ2=(x2(2y)3)โ2
We raise 2y to the indicated power and reorder the expression.
(x2(2y)3)โ2=(x2โ
8y3)โ2=(8x2y3)โ2
Finally we apply the rule of negative exponents.
(8x2y3)โ2=(8x2y3)21โ=64x4y61โ
Answer:
((2y)โ3x2โ)โ2=64x4y61โ
Review questions
What happens when there is a power with a negative exponent?
When we have a power with a negative exponent, we can make the exponent positive by converting the expression to a fraction. First we put a 1 in the numerator, in the denominator we put the original power, but changing the negative sign of the exponent by a positive sign.
When the exponent is negative, is the result is negative?
The negative sign of the exponent does not mean that the result can't be positive. Look at the following example
(3)โ1=31โ
Notice how the sign of the exponent is negative but the result is 1/3 positive.
How do you proceed when the exponent is negative or positive?
If the exponent is positive we just multiply the base by itself as many times as the exponent indicates. If the exponent is negative we first convert the expression to a fraction with positive exponents and proceed as mentioned above.
What is done when the base of a power is negative?
If the base of a power is negative and the exponent is an integer, we are only indicating the sign of the number that will be multiplied by itself, as many times as the exponent indicates. If the exponent is an even number, the sign of the power will be positive, but if it is odd, the sign of the power will be negative.
What happens when the base is negative and the exponent is 0?
If the base is a negative number with zero exponent the result is 1.
Exercises for negative exponents
Exercise 1
Task
7โ24=?
Solution
7โ24=70โ24=
72470โ=7241โ
Answer
7241โ
Do you know what the answer is?
Exercise 2
Task
(8ร9ร5ร3)โ2=
Solution
We will use the formula
(abc)n=anโ
bnโ
cn
8โ2โ
9โ2โ
5โ2โ
3โ2
Answer
8โ2ร9โ2ร5โ2ร3โ2
Exercise 3
Task
((7ร3)2)6+(3โ1)3ร(23)4=
Solution
We will use the formula
(am)n=amโ
n
(7โ
3)2โ
6+3โ1โ
3โ
23โ
4=
2112+3โ3โ
212
Answer
2112+3โ3ร212
Exercise 4
Task
(3ร2ร4ร6)โ4=
Solution
We will use the formula
(aโ
bโ
c)n=anโ
bnโ
cn
(3โ
2โ
4โ
6)โ4=3โ4โ
2โ4โ
4โ4โ
6โ4
Answer
3โ4ร2โ4ร4โ4ร6โ4
Exercise 5
Task
19โ2=?
Solution
19โ2=190โ2
192190โ=1921โ=
3611โ
Answer
3611โ
Do you think you will be able to solve it?
examples with solutions for powers with negative integer exponent
Exercise #1
(41โ)โ1
Video Solution
Step-by-Step Solution
We use the power property for a negative exponent:
aโn=an1โWe will write the fraction in parentheses as a negative power with the help of the previously mentioned power:
41โ=411โ=4โ1We return to the problem, where we obtained:
(41โ)โ1=(4โ1)โ1We continue and use the power property of an exponent raised to another exponent:
(am)n=amโ
nAnd we apply it in the problem:
(4โ1)โ1=4โ1โ
โ1=41=4Therefore, the correct answer is option d.
Answer
Exercise #2
Video Solution
Step-by-Step Solution
We use the property of powers of a negative exponent:
aโn=an1โWe apply it to the problem:
5โ2=521โ=251โ
Therefore, the correct answer is option d.
Answer
Exercise #3
4โ1=?
Video Solution
Step-by-Step Solution
We use the property of raising to a negative exponent:
aโn=an1โWe apply it to the problem:
4โ1=411โ=41โTherefore, the correct answer is option B.
Answer
Exercise #4
7โ24=?
Video Solution
Step-by-Step Solution
We use the property of raising to a negative exponent:
aโn=an1โWe apply it in the problem:
7โ24=7241โTherefore, the correct answer is option D.
Answer
7241โ
Exercise #5
19โ2=?
Video Solution
Step-by-Step Solution
To solve the exercise, we use the property of raising to a negative exponent
aโn=an1โ
We use the property to solve the exercise:
19โ2=1921โ
We can continue and solve the power
1921โ=3611โ
Answer
3611โ