In the world of maths, exponents play the most significant role, as it makes logical and quick approach to the complex methodical problems. Particularly in the method of multiplication, it can be said that exponents have a major role as it actually determines the number of times any numerical value is used in it, thereby it can be easily used for the large and lengthy multiplication by the representation of the number of times the number in particular is used, which is called exponents. Like in case, there is a need to multiply the numerical value 8 five times, so instead of multiplying it five times, the power of five to the number eight can be used, so here the power used is called indices, can in other terms is also called exponents.
The below given illustration which give a clear concept about the exponential terms in mathematics.
Other than using the numeric 8, eight times it can also be written as:
So here, the power 5 is actually the exponents.
Consequently, it can be said that exponents are also called indices.
Example: 63 = 6 × 6 × = 216
In other words it can be called as 63, could be called "6 to the third power", "6 to the power 3" or simply "6 cubed.
Exponents thereby can be regarded the method or the mathematical tool to ease lengthy processes of multiplication.
Example: 34 = 3 × 3 × 3 × 3 = 16
It can stated as 34, could be called "3 to the fourth power" or "3 to the power 4" or simply "3 to the 4th".
So, in the above two examples the two indices used are 3 and 4 respectively which are in other words called the exponents.
The exponential laws which are also termed as the rules of exponents can be easily illustrated by keeping few ideas in mind which will simply the rules to understand and apply them in mathematical problems. Firstly the above said idea of exponents indicating the number of times a particular numeric is used during multiplication. The term exponents is multiplication of the numbers in logical way, whereas the negative exponents means division of numbers. Fractional exponents for example if there is a term like 1/n, which simply indicates to take the nth root. This is a key point which, must be kept in mind while learning exponents and it is used in several mathematical problems to ease the query.
In order to understand the fractional exponents, like 1/n means to take the nth root:
a 1∕n = n√a
The first three laws above (a1 = a, a0 = 1 and a-1 = 1/a) are just part of the natural sequence of exponents.
The law that aman = am+n
Now, it can be explained as how many times do we end up multiplying "a"? The answer would be first "m" times, then by another "n" times, for a total of "m+n" times.
The law that am/an = am-n
Now, this law can be explained as how many times do we end up dividing “a”?
The answer would be first m times, then by another “n” times, for a total of “m-n” times.
The law that (ab)n = anbn
In order to show how this one works, it is needed to think of re-arranging all the "a"s and "b"s as in this example.
Example: (ab)3 = (ab)(ab)(ab) = ababab = aaabbb = (aaa)(bbb) = a3b3
The law that (a/b)n = an/bn
Similarly by taking reference from the previous example, it can be understood that “a"s and "b"s, so we can write it in this manner as well.
Example: (a/b)3 = (a/b)(a/b)(a/b) = (aaa)/(bbb) = a3/b3
The above said laws of exponents are very crucial in solving basic mathematical problems as it can be seen that it is mostly used in the junior standard textbooks. The time taking multiplication can be easily solved depending upon the laws and rules of indices.
The above written law is found be a little more critical and needs time for understanding and applying them ultimately reduces the wastage of time and effort in solving lengthy mathematical sums. As suggested one reads fractional exponents first, or this may not make sense.
Anyway, the important idea is that:
x1/n = The n-th Root of x
And so a fractional exponent like 43/2 is really saying to do a cube (3) and a square root (1/2), in any order.
One must keep in mind that, m/n = m × (1/n):
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