Taking advantage of all the properties of powers or laws of exponents
From time to time, we will come across exercises in which we must use all the properties of powers together. As soon as you have the exercise, try to first get rid of the parentheses according to the properties of powers and then, apply these properties to the corresponding terms, one after the other.
All the properties of powers or laws of exponents are: amรan=a(m+n) anamโ=a(mโn) (aรb)n=anรbn (baโ)n=bnanโ (an)m=a(nโm) a0=1 When a๎ =0 aโn=an1โ
Example of an exercise for which we must use all the properties of powers or laws of exponents:
Example 1
(x3โ)โ3รxโ4(3โ3)4รxโ2โ
We know that this exercise might scare you a lot but, trust us, it includes everything you have already studied. The way to solve it is simply to try to figure out what you can do with the power properties you have learned. Shall we? Observing the exercise, we will see that there are terms that are in parentheses. According to the order of mathematical operations, parentheses are in the first place, so we will start with them. Let's look at the first term with parentheses: (x3โ)โ3 According to this property:ย (baโ)n=bnanโ
It will give us:
(x3โ)โ3=xโ33โ3โ Let's move on to the second term with parentheses: (3โ3)4
According to this property: (an)m=a(nรm)
It will give us:ย (3โ3)4=3โ3=4=3โ12
Great! We have gotten rid of the parentheses! Now we will write the exercise according to what we have reached so far:
xโ33โ3โรxโ43โ12รxโ2โ=
Pay attention that, between our two terms there is a multiplication operation. Therefore, we can join them into a single term by writing them exactly as they appear, but under a fraction bar. Let's not forget the multiplication between them. We will obtain: xโ3รxโ43โ3ร3โ12รxโ2โ We will realize that, in the exercise, there are hidden equal bases with multiplication operation between them. Consequently, we can add the concerning powers. According to the following property:ย amรan=a(m+n)
We will obtain: xโ73โ15รxโ2โ
It's looking much better, right? Let's continue. We will realize that, in the exercise we have now, there is a fraction with equality of bases. Thus, we can subtract the concerning powers. According to the following property: anamโ=a(mโn)
We will obtain: 3โ15รx5=
Pay attention, we subtract the powers โ2 and โ5: When you subtract a negative from a negative, the result is positive.
We have reached the solution, but note that, you might be asked to show the result only with positive powers. We can transform the negative power โ15 to positive according to the following property: aโn=an1โ
We will obtain: 3151โรx5
We will multiply and obtain: 315x5โ
The more you practice using these properties or laws separately, the easier it will be for you to apply them together and know which one to use first in a complex exercise.
Here is an example of a complex exercise that requires the use of several properties together.
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Wow... You might be looking at the exercise and wondering if you have any idea where to start... That's totally natural since it really looks threatening and includes many negative signs, fractions, and parentheses.
But fear not, we are here to solve it together and we will give you recommendations so you can succeed with similar exercises that you will receive from now on, even in exams.
Our first recommendation for this type of exercises is not to torment yourself with fractions and negatives, look at the terms that are in parentheses.
In the order of mathematical operations, this is the first of all, and it is important that we get rid of the parentheses.
Let's start with the first expression and remember the property of the power of a quotient that says that each term is raised to the power separately.
We will realize that in the numerator we already have a power over the base X so we will use the property of the power of a power.
Let's not forget to apply the power also over the coefficient of the X.
We know you still can't see how all this will be solved, but hey, be patient. We have only taken the first step.
Now we will move on to the second term and there we will also remove the parentheses. We will apply the property of power of a power on the X and obtain:
Yโ35โ3รXโ3Xโ6โรY2X8รYโ1ร50โ=
Now we will realize that we have a certain term raised to 0, and we already know that, any term raised to 0 is equal to 1. Consequently, we can ignore it and write the exercise this way:
Yโ35โ3รXโ3Xโ6โรY2X8รYโ1โ=
It's looking a bit more organized now, isn't it?
Let's continue.
Our second recommendation is to combine the fractions, especially if there is multiplication between them. Generally, after removing the parentheses, this will be the next step.
Indeed, now we want to end up with a single fraction. We will notice that there is multiplication between them, so we can do it very easily.
Notice that, when we multiply like bases we can add the exponents and obtain a single base with a single exponent.
We will calculate and obtain:
Yโ15โ3รXโ3X+2รYโ1โ=
Ah! This is already starting to look like something familiar where we can act without fear.
We will realize that, both in the numerator and in the denominator, we have an identical base and exponent that we can reduce. Obviously, we will do it and, in this way, dissolve the fraction.
We will receive:
5โ3รXโ3X+2=
Excellent! Now we will use the property of the negative exponent, we will convert the terms into a fraction according to the rules. Remember that if the negative exponent is an expression we must alter the signs of each of the terms of the expression.
We will receive:
531โรX3Xโ21โ=
We can combine the fractions and arrive at:
125X3Xโ21โ
We're done!
See? Quite simple, isn't it?
The trick is to use the properties you learned and understand which one you can use in each case.
Rest assured that, if you work slowly and carefully, you will reach the correct result.
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Properties of Powers Exercises
Exercise 1
Solve the following exercise:
23ร24+(43)2+2325โ=
Solution:
We start by solving the product: 23ร24
According to the multiplication of powers with the same base, we will add the power coefficients, 3+4
Then the expression (43)2,
According to the power of a power property, we can multiply the powers 2ร3=6.
Finally, in the expression 2325โ we use the property of the quotient of power with the same base, and through this, we subtract the coefficients, 5โ3=2
Therefore
Answer:
22+27+46
NOTE: In this exercise, we can write 4 as 22 and obtain: 22+27+212
In this exercise, there are two central laws, the power of a power property and the power of a product.
Since both are essentially multiplication operations, it is possible to use the commutative law in this case and start by multiplying the coefficients.
3ร4=12
We use the first law also for the second factor of the exercise.
(xรy)
And then we will use the second law to give each of the numbers in parentheses the power, and so it results:
912รx12รy12รz12+ayx
Answer:
912รx12รy12รz12+ayx
Exercise 3
Solve the following exercise:
X3รX2:X5+X4
Solution:
First, we solve the multiplications and divisions from left to right.
Then addition and subtraction
Also, we use the power laws (Power of a product)
X3รX2:X5+X4=
We add the powers that have multiplication with the same base.
According to the power property, when we encounter an expression in which the power value appears throughout the product or in the entire exercise where there are only multiplication operations among the members (using parentheses throughout the expression), we can take the power value and apply it to each product
That is, each of the products is raised to the power.
Therefore 4a9a11a
Answer:
4a9a11a
Exercise 5
Solve the following exercise:
(x2ร3)2=
Solution:
In this task, there is the use of two laws, both the multiplication of powers and the power of a power. Each of the products inside the parentheses receives the external power, since they have different bases and a multiplication operation between them. The power inside the parentheses is multiplied by the power outside of it, according to the law of a power of a power.
The properties of exponents are the laws that tell us how to work with exponents. Depending on the form of the expression or the exercise, we may be able to use some of these laws.
How are the properties of powers applied?
Depending on the exercise, we can apply a property of the exponent. There are properties for multiplication and division of powers with the same base, properties for power of a power, power of products and quotients, as well as the law for zero exponent or negative integer exponent.
We use the power property to multiply terms with identical bases:
amโ an=am+nKeep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:
amโ anโ ak=am+nโ ak=am+n+kWhen we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,
Let's return to the problem:
First keep in mind that:
10=101Keep in mind that all the terms of the multiplication have the same base, so we will use the previous property:
101โ 102โ 10โ4โ 1010=101+2โ4+10=109
Therefore, the correct answer is option c.
Answer
109
Exercise #2
(3ร4ร5)4=
Video Solution
Step-by-Step Solution
We use the power law for multiplication within parentheses:
(xโ y)n=xnโ ynWe apply it to the problem:
(3โ 4โ 5)4=34โ 44โ 54Therefore, the correct answer is option b.
Note:
From the formula of the power property mentioned above, we understand that it refers not only to two terms of the multiplication within parentheses, but also for multiple terms within parentheses.
Answer
344454
Exercise #3
54ร25=
Video Solution
Step-by-Step Solution
To solve this exercise, first we note that 25 is the result of a power and we reduce it to a common base of 5.
25โ=525=52Now, we go back to the initial exercise and solve by adding the powers according to the formula:
anรam=an+m
54ร25=54ร52=54+2=56
Answer
56
Exercise #4
(42)3+(g3)4=
Video Solution
Step-by-Step Solution
We use the formula:
(am)n=amรn
(42)3+(g3)4=42ร3+g3ร4=46+g12
Answer
46+g12
Exercise #5
2324โ=
Video Solution
Step-by-Step Solution
Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:
bnbmโ=bmโnWe apply it in the problem:
2324โ=24โ3=21Remember that any number raised to the 1st power is equal to the number itself, meaning that: