# Index Notation

## Index Notation Definition

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 × 2 × 2.

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

$y^{n} = \frac{y \times y \times y \times y\times . . . \times y \times y}{"n" \text{ "lots of" } "y"}$

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 exponent 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 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:

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

$= 2 \times 2 \times 2 \times 2 \times 2 \times 2$

$= 2^{6} = 2^{(4 + 2)}$

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

Example:

$3^{5} \div 3^{3} = \frac{3\times 3\times 3 \times 3 \times 3}{3\times 3\times 3} = 3^{(5 - 3} = 3^{2} = 9$.

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

Example:

$7^{0} = 1 \text{ or } 8^{0} = 1$

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

Example:

$(2^{2})^{3} = 2^{(2 \times 3)} = 2^{6}$

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

Example:

$a{-x} = \frac{1}{a^{x}}$

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

Example:

$y^{\frac{2}{3}} = (\sqrt[3]{y})^{2}$

### 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:

$6000 = 6 \times 1000 = 6 \times 10^{3}$

6 thousand is 6 times a thousand. And, thousand in 6000 is 10³. Hence 6 times 10³ = 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 $1.988 \times 10^{30}$.

### 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 = $2 \times7 \times 7\times 7\times 7\times 7\times 7\times 7$

Prime factors of 98 in index notation can be represented as $2 \times 7^{2}$

2. Evaluate $(\frac{81}{16})^{\frac{-3}{4}}$

Solution:

$(\frac{81}{16})^{\frac{-3}{4}} = \frac{1}{(\frac{81}{16})^{\frac{3}{4}}}$

$= (\frac{16}{81})^{\frac{3}{4}}$

$= ((\frac{16}{81}) \frac{1}{4})^{3}$

$= (\frac{2}{3})$

$= \frac{8}{27}$

3. Evaluate $2^{3} \times 3^{2} \times 5^{2} \times 3^{3}$. Write the answer in index notational form.

Solution:

$2^{3} \times 3^{2} \times 5^{2} \times 3^{3}$

$= 2^{3} \times 3^{2+3} \times 5^{2}$

$= 2^{3} \times 3^{5} \times 5^{2}$

4. Determine $2^{5} \div 2^{3}$ 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,

$2^{5} \div 2^{3} = 2^{2} = 4$

The answer in index notation can be represented as $2^{2}$.

### 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.

Q1. What are Negative Exponents?

Ans. Negative exponent such as x-n = 1xn says that negative exponents in numerator get moved to the denominator and become positive exponents whereas negative exponents in the denominator get moved to the numerator and become positive. Only the negative exponents are moved. For example,

6-2 = 162 = 136a-3b-7 = b7a3

Q2. Mention the Difference Between Exponents and Power.

Ans. In Mathematics, exponents means the small number, placed at the up-right of the whole number. It represents the number of times the base number is used as a factor in multiplying itself. For example, in 3³ = 3 × 3 × 3, 3 is the exponent which represents the number 3 is to be multiplied by itself.

Power, in Mathematics, refers to the whole number that denotes the repeated multiplication of the same number. For example, in 3³ = 3 × 3 × 3, 3 is the base number that is multiplying itself thrice and read as “ three to the power of three” or “ three to the third power.

Q3. Define Index.

Ans. Index of a variable or (constant) is the value that is raised to the power of variables. The indices are also known as powers or exponents. It states the number of times the given number has to be multiplied. It is represented in the form: 3³ = 3 × 3 × 3.

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