In mathematics while doing arithmetic calculations many times we come across the numbers in the form of powers or exponents then at that time we have to use the laws of exponentials first and then we solve it and finally we get the answer of that question.
So what is mean by exponential?
In mathematics, the function of the form y = ex is called as the exponential function, where e is the base and x is the exponent or power.
For example:
- 23 is also the exponential form where 2 is the base and 3 is the exponent.
- Similarly, 35 is the exponential form where 3 is the base and 5 is the power or exponent.
- Now, when we come across the multiplication or division of two exponentials then what we have to do. So, for that we discuss in detail the laws of exponentials as given below:
1.) am*an = a (m + n)
when we have product of two exponentials with the same base and different exponents then we must write the same base and must take the addition of exponents only as given above.
For example:
- 23*24 = 2(3 + 4) = 27 = 128
Here, the base is same i.e. 2 and we have to take the product of exponentials then we have written the base as it is and taken the sum of powers and finally, we get the answer.
- Similarly, 32*33= 3(2 + 3) = 35= 243
Here, the base is same, and exponents are different and while multiplying them we have written the base as it is and taken the sum of the powers only.
2.) am/ an = am*a-n = a(m – n)
when we have to take division of two exponentials with the same base then initially, we write the exponential in numerator with negative power and then we use the law of multiplication of exponentials.
For example:
- 23/ 22 = 23*2-2 = 2(3 – 2) = 2
Here, there is division of two exponentials. And we have taken the denominator in numerator and by using product rule of exponential we solved the numerical.
- Similarly, we can solve 34/ 32 = 34*3-2 = 3(4 – 2) = 32 = 9
- In this way we can solve the division of two exponentials.
3.) am*bm = (ab)m
Here, the bases are different, but their exponents are the same in product of two exponentials. In such case we take the power in common and multiply the bases first and we take its power which is in common.
For example:
- 23*33= (2*3)3 = 63= 216
Here, we have product of two exponentials with the different bases but the same powers. Then we taken product of bases and power is in common and we found the answer.
- Similarly, we can solve 32*42= (3*4)2=122 = 144
Here also the bases are different, and powers are the same so we used the laws of exponentials to solve it.
4.) am/bm = (a/b)m
Here, there is a division of two exponentials with the different bases and the same exponent. So we have taken their power in common and solved it.
For example:
- 83/ 43= (8/4)3 = (2)3 = 8
Here we have taken the two exponentials with the different bases and power 3 is in common and hence we have taken the power in common and solved the ratio and finally found the answer.
- Similarly, we can solve 92/ 32 = (9/3)2 = 32 = 9
In this example also there is division of two exponentials with same powers and different bases, so we have taken out their power common and solved it.
5.) (am)n = amn
Here, there is an exponential whose again there is power so by using laws of exponential we take multiplication of powers and the base remains same.
For example:
- (23)4 = 23*4= 212 = 4096
- Here also there is an exponential in the bracket and its whole power is again 4, so we have taken the product of powers and then solved the example.
- Similarly, we can solve (32)2= 32*2= 34= 81
- Here also, there is power of exponential so we can take multiplication of powers and then solved it.
Note:
- This property of exponential is used only when there is a power of whole exponential and not the power of exponent only.
- That means, this property is used when (am)n
6.) a1/m = m√a
here, the power is 1/m which means mth root of a.
For example:
- 641/3 = 3√64 = 4
Here, power of 64 is 1/3 which means cube root of 64.
- Similarly, 1251/3 = 125 =5
Here also power of 125 is 1/3 which means that cube root of 125.
7.) a-m = 1/am
when there is negative sign in power then we use this rule.
For example:
- 4-2 = 1/42 = 1/16
- Similarly, 3-3 = 1/33 = 1/27
8.) am/n = n√am
Here, the power of a is m/n which means that nth root of am.
For example:
43/2 = √43 = √64 = 8
Here, power of 4 is 3/2 which means that square root of 43.
Note:
When power or exponent of any variable or number is zero then its value is always one.
For example:
X0 = 1 and 20= 1, 350 = 1
Tower rule: