Law of Exponents

If I would have to describe Exponents in a simple way, I would say that essentially they are the power to which a number is raised. So if we square a number, it would mean that the exponent of the number is 2 and similarly if we were to cube the number, its exponent would be 3, and so on. The number to which the power is raised is called the base or the base number and the power in itself is called the exponent. This chapter is very interesting because we are going to learn The Laws of Exponents which will help us in solving the problems very easily. 

Laws of Exponents

• In the first law, we will see what happens when we try to multiply two same base numbers with different powers or exponents.

Let us assume one number am and another number an

(Note: It is the same number a with different powers m and n)

So, the first law states that if we try to multiply these two numbers, their powers will be added.

am * an = a (m+n)

Let us take an example,

Let us try to multiply 77 and 73

So,

77 * 73 = 7 (7+3) = 710 = 282475249

The first law of exponents states that if multiply two same base numbers but with different powers, their powers will be added.

• Now we will see what happens when we try to divide two same base numbers but with different powers.

Let us assume one number am and another number an

(Note: It is the same number a with different powers m and n)

The second law states that if we try to divide the same base number with different powers, the final result will be the same base number raised to the power which is a result of the initial two powers being subtracted from each other.

So,

(am/ an) = a(m-n)

Let us proceed with an example,

Let us take 76 and 73,

Now trying to divide them by each other,

(76/73)= (73) =343

So to express the second law in words, we can say that if we try to divide the same base number with different powers, the final result will be the same base number raised to the power which is a result of the initial two powers being subtracted from each other.

• Now let us see what happens when we try to raise a base number which has been raised to a particular power to another power.

Consider a number am,

Now we aspire to raise am to another power n,

We have to find (am) n

The third law states that if you wish to raise a number which has been raised to a particular power, to another power, the final result will be that base number raised to a power which is a product of those two initial powers.

So (am) n = a(m*n)

(Note: if m and n are the same power (am) m= a(m*m) )

Let us demonstrate this law with an example.

Take the number 74, now we will raise 74 to the power of 3

So final result,

(74)3= (712) =13841287200

So to summarize this law in words, this law states that if you wish to raise a number which has been raised to a particular power, to another power, the final result will be that base number raised to a power which is a product of those two initial powers.

• Now let us see what will happen when we try to multiply two numbers and raise them to a single power.

Let us consider two numbers a and b and a power m

We need to find the result of (a*b) m

The fourth law states that if we try to multiply two numbers and raise them to a single power, the result will be the product of each of those numbers individually raised to that power.

So let us demonstrate this law with an example,

Take two numbers 5 and 7,

Now we need to find (5*7)3

Solving it using the law,

(5*7)3 = (53) * (73) = 42875

So to summarize this law using words, we can say that we try to multiply two numbers and raise them to a single power; the result will be the product of each of those numbers individually raised to that power.

• Now we will consider what happens when we try to divide two different base numbers and raise them to a single power.

Let us consider two numbers a ad b and a power m,

We aspire to find the result of (a/b)m

The fifth law states that if we wish to divide two different base numbers and raise them to a single power, the result of such an operation would be the division of the two numbers individually raised to that power.

So to demonstrate this with an example,

Let us take two numbers 7 and 5 and the power 3,

We need to find, (7/5)3

Using the law,

(7/5)3 = (73/53) = 2.744

So to summarize in words, this law states that if we wish to divide two different base numbers and raise them to a single power, the result of such an operation would be the division of the two numbers individually raised to that power.

Solved Examples

Q1. Find the result of 32 * 34.

Soln. The solution will be,

32 * 34 = 36 = 729

Q2. Find the result of (54/53).

Soln. The solution will be,

(54/53) = (51) = 5

Q3. Find the result of (22)4.

Soln. The solution will be,

(22)4 = (28) = 256

Q4. Find the result of (7/6)2.

Soln. The solution will be,

(7/6)2 = (72/62) = (49/36) = 1.361

Q5. Find the result of (4*5)2.

Soln. The solution will be,

(4*5)2 = 42 * 52 = 16 * 25 = 400

Uses of Laws of Exponents

The Laws of Exponents are mostly used to solve the complex mathematical problems which use complex exponential expressions and mathematical expressions which use operations like multiplication and division multiple times.

Conclusion

In this chapter first we discussed the law of exponents. Then we moved on to the introduction part where we saw the Laws of Exponents and we solved quite a few problems in that part. Then we saw a few solved examples pertaining to each Law of the Exponents. Then we finally saw the uses of The Laws of Exponents.