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 \[2 \times 2 \times 2 \].
\[3^{2}\] 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:
\[ Y\frac{2}{3} = (\sqrt[3]{Y})^{2}\]
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:
\[6000=6*1000= 6* 10^{3}\]
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.
FAQs on Index Notation
1. What is index notation in Mathematics?
Index notation is a method used to write numbers that are multiplied by themselves multiple times in a shortened form. It consists of two parts: a base (the number being multiplied) and an index or exponent (the small number that indicates how many times the base is multiplied). For example, instead of writing 5 × 5 × 5, we can use index notation to write it as 5³, where 5 is the base and 3 is the index.
2. How can you express a number like 32 using index notation?
To express a number using index notation, you first need to find its prime factors. For the number 32, the prime factorization is 2 × 2 × 2 × 2 × 2. Since the number 2 is being multiplied by itself 5 times, we can write this in index notation as 2⁵. Here, 2 is the base and 5 is the index or exponent.
3. What is the difference between an 'exponent' and a 'power'?
While often used interchangeably, 'exponent' and 'power' have distinct meanings. The exponent (or index) is the small superscript number that tells you how many times to multiply the base by itself. The power refers to the entire expression (like 5³) or the result of the calculation (125). So, in the expression 5³, the exponent is 3, and we say '5 is raised to the power of 3'.
4. What are the fundamental laws of index notation for the 2025-26 syllabus?
The laws of index notation (or laws of exponents) are rules that help simplify expressions. The key laws as per the NCERT syllabus are:
- Multiplication Law: When multiplying terms with the same base, you add the exponents. Example: aᵐ × aⁿ = aᵐ⁺ⁿ
- Division Law: When dividing terms with the same base, you subtract the exponents. Example: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Power of a Power Law: When a term with an exponent is raised to another exponent, you multiply them. Example: (aᵐ)ⁿ = aᵐⁿ
- Zero Exponent Law: Any non-zero base raised to the power of zero equals 1. Example: a⁰ = 1
5. What do negative exponents mean in index notation?
A negative exponent indicates a reciprocal or repeated division. If you have a base raised to a negative exponent, like x⁻ⁿ, it is equivalent to 1 divided by that base raised to the positive exponent, i.e., 1/xⁿ. For example, 4⁻² does not mean -16; it means 1/4², which equals 1/16. It essentially moves the base from the numerator to the denominator (or vice versa) and makes the exponent positive.
6. Why does any non-zero number raised to the power of zero equal one?
This rule can be understood using the division law of indices. Consider the expression a³ ÷ a³. We know that any number divided by itself is 1. Using the division law, we subtract the exponents: a³ ÷ a³ = a³⁻³ = a⁰. Since both expressions are equal, it proves that a⁰ = 1. This logic holds for any non-zero base 'a'.
7. How do fractional indices relate to roots?
Fractional indices are a way of representing roots of a number. In a fractional index like xᵃ/ᵇ, the denominator 'b' represents the root to be taken, and the numerator 'a' represents the power. For example, 64¹/³ means the cube root of 64, which is 4. Similarly, 25³/² means the square root of 25 (which is 5), raised to the power of 3, resulting in 5³ or 125.
8. What is a common mistake students make when adding terms with exponents?
A very common mistake is to incorrectly apply the multiplication law to addition. For example, students might think x² + x³ is equal to x⁵. This is incorrect. The laws of indices only apply when multiplying or dividing terms with the same base. Terms like x² + x³ are considered separate and cannot be simplified by adding their exponents. They can only be added if they are like terms (e.g., 2x² + 3x² = 5x²).
9. Where is index notation used in real-world applications?
Index notation is crucial in many fields beyond the classroom. For instance:
- In science, it's used in scientific notation to express very large numbers (like the distance to stars) or very small numbers (like the size of an atom).
- In computer science, memory and storage are measured in powers of 2 (e.g., 1 Kilobyte = 2¹⁰ bytes, 1 Megabyte = 2²⁰ bytes).
- In finance, the formula for compound interest uses exponents to calculate growth over time.
