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Application of exponents in real world One of the application is to measure the strength of earthquakes on the Richter scale. A level 1 earthquake is 1 × 101, a level 2 earthquake is 1 × 102, and a level 7 earthquake is 1 × 107, etc.
Exponents

A numerical expression is a mathematical phrase that contains only numbers and mathematical operations. For example, “6 ÷ 3 – 2 + 1” is a numerical expression. In an expression involving multiplication, the quantities multiplied are called factors. For example, in an expression: “3 × 7 × 11 × 13”, 3, 7, 11 and 13 are called factors of an expression.

Exponents are used to represent repeated factors in multiplication. For example, the expression 64 represents the number that we obtain when 6 is used as a factor 4 times. The number 6 is called the base (sometimes as radix), the number 4 is called the exponent (or index) and the expression 64 is called a power. The exponent in a power represents the number of times the base is used as a factor.

Laws of Exponents

Let ‘a’, ‘b’ be any two non–zero real numbers and ‘m’, ‘n’ be any two positive integers. Then, we have the following laws:

i) The product law:

am × an = am + n
The product law states that when multiplying powers with the same base, keep the base and add the exponents.

ii) The quotient law:


The quotient law states that when dividing powers with the same base, keep the base and subtract the exponents.

iii) The power of a power law:

(am)n = a(m × n)
The power of a power law states that when we have a power of a power, keep the base and multiply the exponents.

iv) The power of a product law:

(ab)m = am × bm
The power of a product law states that when we have a power of a whole multiplication, keep the base and multiply the exponent of the product with each of the exponents of the factors.

v) The power of a quotient law:


The power of a quotient law states that when we have a power of a whole division, keep the base and multiply the exponent of quotient by the exponent of numerator and by the exponent of denominator.

vi) The zero exponent law:

a0 = 1, a ≠ 0
The zero exponent law states that any number (except 0) to the power of 0 is 1.

vii) The negative exponent law:


The negative exponent law states that any number raised to negative exponent is the same as finding the reciprocal of the same number raised to the positive base.

Summary of Laws   Example   Brain Teaser  

Fractional Indices

What is the meaning of a1/2?
To find out, let us multiply it by itself.
a1/2 × a1/2 = a1/2 + 1/2 = a1 = a.
∴ a1/2 is such a number when multiplied by itself yields the product 'a'. By arithmetical definition, such a number is the square root of 'a'.
∴ a1/2 = √a
Ex: 31/2 = √3 = 1.732; (16)1/2 = √(16) = 4, etc.

In general, if 'n' is any positive integer, we may conclude:
a1/n = (nth root of 'a').
Similarly, we can deduce that, if 'm' and 'n' are any positive integers,
am/n = .
Ex: a2/5 = .

Indices which are in decimal form can be changed to fractions.
Ex: a0.25 = a1/4 = .

If 'a' is a negative real number and 'n' is an even positive integer, then a1/n is not defined. Thus, (-3)1/2 is not defined.

For positive value of 'a', the value of a1/n will always be taken as positive.

Conclusion: Refer to adjacent graph of y = 2x. Any number (within the limits of the plot) can be expressed as a power of 2. Conversely, any number can be used as index of some power of 2.

Order of Operations – BODMAS or PEMDAS

Some numerical expressions have more than one mathematical operation. The value of this type of numerical expression depends on the order in which the operations are evaluated. To find the correct value of the expression, follow the order of operations as under:

(i) Do all operations that occur within grouping symbols, such as parentheses () , brackets [] , and braces {}.
(ii) Evaluate all powers in order from left to right.
(iii) Do all multiplications and/or divisions from left to right.
(iv) Do all additions and/or subtractions from left to right.
Note: PEMDAS is a mnemonic to help students remember the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

BODMAS is also another mnemonic standing for Brackets, Order (Exponent), Division, Multiplication, Addition, Subtraction.

Removal of Brackets: We strictly remove different types of brackets in the following order:
(i) Bar or Vinculum (—)
(ii) Parentheses ()
(iii) Curly Brackets {}
(iv) Square Brackets [].

Ex: Evaluate [24 ÷ {12 – (10 – 6 – 2)}].
Sol:
[24 ÷ {12 – (10 – 6 – 2)}] = [24 ÷ {12 – (10 – 4)}] [∵ Removing 'Bar']
= [24 ÷ {12 – 6}] [∵ Removing 'Parentheses']
= [24 ÷ 6] [∵ Removing 'Curly Brackets']
= 4 [∵ Removing 'Square Brackets']