What Is Index Notation Definition Rules and Solved Examples
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 and Laws of Indices Explained
1. What is index notation in Maths?
Index notation is a way of writing repeated multiplication of the same number using powers (exponents).
For example:
- 2³ means 2 × 2 × 2
- Here, 2 is the base
- 3 is the index (exponent or power)
Index notation makes expressions shorter and easier to work with in algebra and arithmetic.
2. What are the laws of indices?
The laws of indices are rules that simplify expressions with powers of the same base.
- am × an = am+n
- am ÷ an = am−n (a ≠ 0)
- (am)n = amn
- a0 = 1 (a ≠ 0)
- a−n = 1/an
These exponent rules are essential for simplifying algebraic expressions and solving equations.
3. How do you multiply numbers using index notation?
To multiply powers with the same base, add the indices.
Use the rule: am × an = am+n.
Example:
- 3² × 3⁴ = 32+4 = 3⁶
- 3⁶ = 729
This rule only works when the base numbers are the same.
4. How do you divide numbers with indices?
To divide powers with the same base, subtract the indices.
Use the rule: am ÷ an = am−n.
Example:
- 5⁷ ÷ 5³ = 57−3 = 5⁴
- 5⁴ = 625
This exponent law applies only when the bases are identical.
5. What does a negative index mean?
A negative index means you take the reciprocal of the base raised to the positive power.
The rule is: a−n = 1/an (a ≠ 0).
Example:
- 2−3 = 1/2³ = 1/8
Negative exponents do not make the number negative; they represent division.
6. What is the value of any number to the power of zero?
Any non-zero number raised to the power of zero equals 1.
The rule is: a0 = 1 for a ≠ 0.
Example:
- 7⁰ = 1
- 100⁰ = 1
This rule comes from the division law of indices.
7. How do you raise a power to a power?
To raise a power to a power, multiply the indices.
Use the rule: (am)n = amn.
Example:
- (4²)³ = 42×3 = 4⁶
This law simplifies expressions involving brackets and exponents.
8. What is fractional index notation?
A fractional index represents a root, where the denominator is the root and the numerator is the power.
The rule is: am/n = (ⁿ√a)m.
Example:
- 81/3 = ³√8 = 2
- 163/4 = (⁴√16)³ = 2³ = 8
Fractional indices connect powers and roots in algebra.
9. What is the difference between index notation and standard form?
Index notation expresses repeated multiplication, while standard form expresses very large or small numbers using powers of 10.
- Index notation example: 5⁴
- Standard form example: 4.2 × 10³
Standard form is a special use of index notation specifically with base 10 in scientific notation.
10. What are common mistakes when using index laws?
Common mistakes in index notation usually involve misapplying the exponent rules.
- Adding indices when bases are different (e.g., 2² × 3²)
- Forgetting that a0 = 1
- Thinking a−n makes a negative number
- Adding instead of multiplying in (am)n
Carefully applying the correct law of indices prevents most algebra errors.
