[Maths Class Notes] on Index Notation Pdf for Exam

Index notation is a method of representing numbers and letters that have been multiplied by themself multiple times.

For example, the number 360 can be written as either [2 times 2 times 2 times 3 times 3 times 5] or [2^{3} times 3^{3} times 5]. 

[2^{3}] is read as ‘’2 to the power of 3” or “2 cubed” and means [2 times 2 times 2 ].

[3^{2}] is read as ‘’3 to the power of 2” or “3 squared” and means [3 times 3 ].

[ Y^{n}  = frac{y*y*y*y*…*y*y}{“n”  “lots of”  “y”}]

In general, yⁿ is read as “y to the power of n” and means “n lots of y”, multiplied together”.

Numbers represented in index notation are often known as exponents or powers. In the above notation, y is the base number, and n is the exponent.

What is Known as an Index Number? 

An index number is defined as the number which is raised to the power. The power says how many times the number is to be used in multiplication.

Generally, it is represented as a small number to the right side and above the base number.

In the above example, the little “3” says to use 8 three times in multiplication. It is read as “ 8 to the power of 3”.

Index Notation Rules

Following are some of the exponent or index rules. These are basic rules of:

Rule 1: When two numbers with the same base are multiplied, their powers get added.

Example:

[ 2^{4} * 2^{2} =(2*2*2*2)(2*2)]

[= (2*2*2*2*2*2)]

[ = 26 = 2^{(4+2)}]

Rule 2: When two numbers with the same base are divided, their powers get subtracted.

Example:

[ frac{3^{5}}{3^{3}}= frac{3*3*3*3*3}{3*3*3} = 3^{(5-3)} = 3^{2} = 9 ]

Rule 3: Any number raised to 0 is equal to 1.

Example:

70=1 or 80=1

Rule 4: If any term with power is raised to the exponent or power, the exponents or powers are multiplied together.

Example:

[ (2^{2}) ^{2} = 2^{2*3} = 26]

Rule 5: Any negative powers can be represented in a fractional form.

Example:

[a – x = frac{1}{ax}]

Rule 6: The exponent or index given in a fraction form can be represented as the radical form.

Example:

[ Yfrac{2}{3} = (sqrt[3]{Y})^{2}]

Power of 10

Power of 10 is a unique way of writing large numbers or smaller numbers. Instead of using so many zeroes, you can show how many powers of the 10 will make that many zeroes. For example, 6000 in the power of 10 can be written as:

[6000=6*1000= 6* 10^{3}]

6 thousand is 6 times a thousand. And, a thousand in 6000 is [ 10^{3}]. Hence 6 times [ 10^{3} = 6000].

Power of 10 is extremely used by Scientists and Engineers as they deal with the numbers that include large numbers of zeroes. For example, the mass of the Sun  that is 1988,000,000,000,000,000,000,000,000 kgs can be written in power of 10 as 1.988×1030. 

Index Notation Examples

Following are some of the index notation examples:

1. Express the prime factors of 98 in index notation form.

Solution:

Prime factors of 98 are = 2×7×7×7×7×7×7×7

Prime factors of 98 in index notation can be represented as 2×72

2. Evaluate [ frac{81}{16}^{-frac{3}{4}}]

Solution: 

[ frac{81}{16}^{-frac{3}{4}}]

[ = frac{1}{frac{81}{16}^{frac{3}{4}}}]

[ = (frac{16}{81})^{frac{3}{4}}]

[ = frac{16}{81}(frac{1}{4})^{3}]

[ = frac{2}{3}^{3}]

[ = frac{8}{27}]

3. Evaluate [2^{3}*3^{2}*5^{2}*3^{3}]. Write the answer in index notational form.

Solution:

[2^{3}*3^{2}*5^{2}*3^{3}]

[ = 2^{3}*3^{2+3}*5^{2}]

[ = 2^{3}*3^{5}*5^{2}]

4. Determine 25÷23 and express the answers in index notation.

Solution: As we know, when two numbers with the same base are divided, their powers get subtracted.

Accordingly,

25÷23

=  22

=  4

The answer in index notation can be represented as 22. 

Did You Know? 

The distance light travels in one year can be easily calculated in the form of index notation as 9.461 × 10¹⁵.

Index Notation is also known as exponential form or exponential notation.