Exponents

We know that multiplication of numbers can be visualized as repeated addition of the same number.
For example, We can visualize 76 multiplied by 87 as the number ’76’ added 87 times to itself, or the number ’87’ added 76 times to itself. The numerical multiplication is therefore called ‘commutative‘ (a times b = b times a; a and b both can be any number).

Similarly, exponentiation or ‘raising a number to another number’ is simply repeated multiplication. For example, when we say ‘2 raised to power 4’ or ‘2 raised to 4’ or sometimes it is said that ‘2 to the 4’, we are supposed to multiply 2 four times with itself.

When we have to find the numerical value of 2 raised to 4 (written as 2^4), we will write it as 2 x 2 x 2 x 2.
Let’s work out this simple example:
2 x 2 x 2 x 2 = 2 x 2 x (2 x 2)
= 2 x (2 x 4)
= 2 x 8
= 16

Important fact to remember: In the operation $$2^4$$ 2 is called as the ‘base’ and 4 is called as the ‘power’ or the ‘exponent’.

Use of the exponentiation operation in mathematics is to calculate very large numbers in mathematical equations. It is also used to multiply and divide extremely large numbers in a very efficient way by applying the ‘rules of exponents’. Many numbers like Moser’s number, Graham’s number, Skewes’ number are formed using this very operation.

The real life applications of exponents are:
1. Calculate the area of regular polygons (generally any number is raised to power 2 (squared) as a regular polygon has all sides equal)
2. Calculate the volume of regular solids (e.g. Sphere, Cube, Tetrahedron, Pyramid, etc.) (in this case, the length of side is raised to power 3 as all regular solids have same dimensions in all three directions)
3. Calculate very great scale of numbers and basically to calculate volume of n – spatial dimensional cube (e.g. volume of a 4d spatial cube of side 2 cm is
2^4 = 16 cm^4)

Rules of exponents:

$$ Rule \space number \space 1: a^m \cdot a^n = a^{m+n} $$

$$ Example: 2^3 \cdot 2^4 = 2^{3+4} = 2^7 $$

$$ (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 8 \times 16 = 128 $$

The above rule simply means that if you multiply any number raised to m with the same number raised to n, their exponents simply add together to yield the above explained result.

$$ Rule \space number \space 2: \frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0) $$

$$ Example: (\frac{2^5}{2^3}) = 2^{5-3} = 2^2 $$

$$ (\frac {2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2}) = 2^2 = 4 $$

The above rule simply means that if you divide any number raised to m with the same number raised to n, their exponents simply subtract to yield the above explained result.

$$ Rule \space number \space 3: a^m \cdot b^m = (a \cdot b)^m $$

$$ Example: 2^3 \cdot 3^3 = (2 \times 3)^3 $$

$$ ({2 \times 2 \times 2} \times {3 \times 3 \times 3}) = 6^3 or (8 \times 27) = 216 $$

When the bases are different but the exponent is the same, we can simply multiply all the bases and raise their product to the exponent given in the problem.

$$ Rule \space number \space 4: (a^m)^n = a^{m \cdot n} $$

$$ Example: (2^3)^2 = 2^{3 \cdot 2} = 2^6 $$

$$ (2^3)^2 = 8^2 = 64 $$

The exponents can be multiplied when we have a compound exponentiation, i.e. When the above exampled is to be worked out, we use the rule of solving the expression in the bracket first and then raise the number to another exponent written outside the bracket.

$$ Rule \space number \space 5: \frac {a^m}{b^m} = (\frac {a}{b})^m $$

$$ Example: (\frac {6^2}{3^2}) = (\frac {6}{3})^2 $$

$$ (\frac {6}{3})^2 = 2^2 = 4 $$

When the numerator and the denominator have the same exponent, we can perform the division first and raise the quotient to the exponent mentioned in the problem.

$$ Rule \space number \space 6: a^1 = a $$

Any number raised to one is equal to not multiplying the number at all. That is why any number raised to one remains exactly the same.

$$ Rule \space number \space 7: a^0 = 1 $$

Any number raised to zero will give the value as one. The detailed reason might be explained in the Logarithms, but as of now try to convince that raising a number to 0 will yield the value 1.

$$ Rule \space number \space 8: a^{-m} = (1/a^m) $$

$$ Example: 2^{-4} = (1/2^4) $$

$$ 2^{-4} = (1/2^4) = 1/16 $$

When exponents are use in reciprocals, they simply change their signs. The value of exponentiating a number does not change.

$$ Rule \space number \space 9: a^{\frac {m}{n}} = (\sqrt [n] {a^m}) $$

$$ Example: 2^{\frac {4}{2}} = (\sqrt [2] {2^4}) $$

$$ (\sqrt [2] {2^4}) = \sqrt 16 = 4 $$

Just like raising a number to an already raised number will give you the value of raising the number to the product of two exponents, raising a number to the quotient of two exponents will give you the nth root of a number raised to power m.

Now that we have learned the basic rules for exponents, one should practice and master them by writing and working out with different numbers and examples as it is very important to go beyond the simplicity of the numbers like 2 and 3. As the difficulty level goes up, the numbers are replaced by decimal numbers or functions, or composite functions. Regardless of the entity that is in the problem, the rules of exponent remain the same, and are to be applied exactly the same way in all situations.

All the best learning this important mathematical exercise for making your math skills stronger. Feel free to reach out for any doubts or questions.