Dividing exponents with different bases and different powers

Video transcript

- [Narrator] Let's get some practice with our exponent properties, especially when we have integer exponents. So, let's think about what four to the negative three times four to the fifth power is going to be equal to. And I encourage you to pause the video and think about it on your own. Well there's a couple of ways to do this. See look, I'm multiplying two things that have the same base, so this is going to be that base, four. And then I add the exponents. Four to the negative three plus five power which is equal to four to the second power. And that's just a straight forward exponent property, but you can also think about why does that actually make sense. Four to the negative 3 power, that is one over four to the third power, or you could view that as one over four times four times four. And then four to the fifth, that's five fours being multiplied together. So it's times four times four times four times four times four. And so notice, when you multiply this out, you're going to have five fours in the numerator and three fours in the denominator. And so, three of these in the denominator are going to cancel out with three of these in the numerator. And so you're going to be left with five minus three, or negative three plus five fours. So this four times four is the same thing as four squared. Now let's do one with variables. So let's say that you have A to the negative fourth power times A to the, let's say, A squared. What is that going to be? Well once again, you have the same base, in this case it's A, and so since I'm multiplying them, you can just add the exponents. So it's going to be A to the negative four plus two power. Which is equal to A to the negative two power. And once again, it should make sense. This right over here, that is one over A times A times A times A and then this is times A times A, so that cancels with that, that cancels with that, and you're still left with one over A times A, which is the same thing as A to the negative two power. Now, let's do it with some quotients. So, what if I were to ask you, what is 12 to the negative seven divided by 12 to the negative five power? Well, when you're dividing, you subtract exponents if you have the same base. So, this is going to be equal to 12 to the negative seven minus negative five power. You're subtracting the bottom exponent and so, this is going to be equal to 12 to the, subtracting a negative is the same thing as adding the positive, twelve to the negative two power. And once again, we just have to think about, why does this actually make sense? Well, you could actually rewrite this. 12 to the negative seven divided by 12 to the negative five, that's the same thing as 12 to the negative seven times 12 to the fifth power. If we take the reciprocal of this right over here, you would make exponent positive and then you would get exactly what we were doing in those previous examples with products. And so, let's just do one more with variables for good measure. Let's say I have X to the negative twentieth power divided by X to the fifth power. Well once again, we have the same base and we're taking a quotient. So, this is going to be X to the negative 20 minus five cause we have this one right over here in the denominator. So, this is going to be equal to X to the negative twenty-fifth power. And once again, you could view our original expression as X to the negative twentieth and having an X to the fifth in the denominator dividing by X to the fifth is the same thing as multiplying by X to the negative five. So here you just add the exponents and once again you would get X to the negative twenty-fifth power.

Exponents and powers are used to simplify the representation of very large or very small numbers. Power is a number or expression that represents the repeated multiplication of the same number or factor. The value of the exponent is the number of times the base is multiplied by itself.

Example for exponents, If we need to express 3 × 3 × 3 × 3 × 3 in a simple way, we may write it as 35, where 3 is the base and 5 is the exponent. The entire expression 35 is considered to represent power. Example for powers:53 = 5 raised to power 3 = 5 × 5 × 5 = 125, 64 = 6 raised to power 4 = 6 × 6 × 6 × 6 = 1296. A number’s exponent represents the number of times the number has been multiplied by itself. 3 is multiplied by itself for n times, 3 × 3 × 3 × 3 × …n times = 3n. 3n is an abbreviation for 3 raised to the power of n. As a result, exponents are sometimes known as power or, in certain cases, indices.

General Form of Exponents 

The exponent indicates how many times a number should be multiplied by itself to obtain the desired results. As a result, any number ‘b’ raised to the power ‘p’ may be expressed as :

bp =  {b × b × b × b × …  × b} p times

Here b is any number, and p is a natural number.

• Here, bp is also called the pth power of b.
• ‘b’ represents the base, and ‘p’ is the exponent or power.
• Here ‘b’ is multiplied ‘p’ times, and thereby exponentiation is the simplified method of repeated multiplication.

Some basic rules of Exponents 

1. Product Rule ⇢ an × am = an + m
2. Quotient Rule ⇢ an / am = an – m
3. Power Rule ⇢ (an)m = an × m or m√an = an/m
4. Negative Exponent Rule ⇢ a-m = 1/am
5. Zero Rule ⇢ a0 = 1
6. One Rule ⇢ a1 = a

How to Multiply and Divide Exponents?

Solution: 

Exponents and powers are used to simplify the representation of very large or very small numbers.

To Divide Exponents

The laws of exponents simplify the process of simplifying expressions. When dividing exponents with the same base, the basic rule is to subtract the given powers. This is also known as the Division Law or  Exponent Quotient Property.

mn1 ÷ mn2 = mn1/ mn2 = m(n1 – n2)

First Case: Dividing Exponents with the Same Base 

We utilize the basic rule of subtracting the powers to divide exponents with the same base. Consider the expression mn1 ÷ mn2, where  ‘m’ is the common base and the exponents ‘n1‘ and ‘n2‘ are the exponents. According to the ‘Quotient property of Exponents,’

mn1 ÷ mn2 = mn1/ mn2= m(n1 – n2)  

Example: Divide 35 ÷ 33 

Here as we can see bases are same but different powers .

So the division law or Quotient law  : mn1 ÷ mn2   =  mn1/ mn2 = m (n1 – n2)  

Here, 35 ÷ 33

= 35/33

= 3(5-3)

= 32   

Second Case: Dividing Exponents with different Bases 

We apply the ‘Power of quotient property’ to divide exponents with different bases and the same exponent, which is

(m/n)p = mp/np  

Consider the formulas mp ÷ np, which has distinct bases but the same exponent.

Example: Divide: 153 ÷ 33.

This can be solved using the ‘Power of quotient property’ as,

(m/n)p = mp/np.

= 153 ÷ 33

= (15 / 3)3

= 53.    

To Multiply Exponents 

First Case: When Multiply exponents with the same Base 

According to this rule: The product of two exponents with the same base but distinct powers equals the base raised to the sum of the two powers or integers; this is also known as the Multiplication Law of Exponents. When multiplying two expressions with the same base, we can use, 

mn1 × mn2 = m(n1 + n2)  

Where m is the common base and n1 and n2 are the exponents.

For Example, Multiply 33 × 36?

Given: 33 × 36 

Here bases are same. So we will use: mn1 × mn2 = m(n1 + n2)  

Therefore, = 3 (3+6)

= 39 

Second Case: When Multiply Exponents with a different bases

When there is different base with same exponents , we will use the formula : 

mp × np = (m × n)p. 

Here m and n are the different bases and  p is the exponent.

Example: Multiply 23 × 43

Given: 23 × 43 

Here, we will use: mp × np = (m × n)p

= (2 × 4)3

= 83               

In these ways in different cases we can divide and multiply Exponents.

Sample Questions

Question 1: Simplify or Divide 254/54  

Solution: 

Here bases are different with same Exponent,

We will use the formula, (m/n)p = mp/np  

Therefore, = 254/54

= (25/5)4

= 54

= 625

Question 2: Find the value of the expression, 158 × 153

Solution:

Given: 158 × 153

When multiplying two expressions with the same base but different exponent, 

mn1 x mn2 = m(n1 + n2) formula, where m is the common base and n1 and n2 are the exponents.

By Applying this rule, 

we get, = 158  × 153

= 15(8 + 3)

= 1511

Question 3: What is the product of (2x3y5 ) and (3x4y2)?

Solution:

The product of  (2x3y5) and (3x4y2)

= (2x3y5) × (3x4y2)

= (2 × 3) × x3x4 × y5y2                    

When multiplying two expressions with the same base, we can use mn1 × mn2 = m(n1 + n2) formula, where m is the common base and n1 and n2 are the exponents.

= 6x3+4 × y5+2

= 6x7y7 

Question 4: What is x3 divided by x2?

Solution:

Here given: x3divided by x2 

here bases are same but exponents are different,

So we use the division law or Quotient law: mn1 ÷ mn2   =  mn1/ mn2 = m (n1 – n2)  

So write it as x3/x2

= x3 – 2

= x1

= x

Question 5: Evaluate a3 × a5 × a-6 

Solution:

Given that: a3 × a5 × a-6 

Here bases are same but exponents are different ,By using product rule or multiplication law .

mn1× mn2 = m(n1 + n2)

= a3 × a5 × a-6

= a(3 +5) × a-6

= a8 × a-6

= a{8+ (-6)} {Using by product rule}

= a8-6

= a2 

Question 6: Divide 105/55

Solution:

Here bases are different with same Exponent ,

we will use the formula  : (m/n)p = mp/np  

Therefore, = 105/55

= (10/5)5

= 55

= 3125

How do you divide exponents with different bases and negative powers?

What do you do when you have different bases and different exponents?

How do you divide exponents with different variables?