Index
Index (indices) in Maths is the power or exponent which is raised to a number or a variable. For example, in number 2^{4}, 4 is the index of 2. The plural form of index is indices. In algebra, we come across constants and variables. The constant is a value which cannot be changed. Whereas a variable quantity can be assigned any number or we can say its value can be changed. In algebra, we deal with indices in terms of numbers. Let us learn the laws/rules of the indices along with formulas and solved examples.
Index Definition
A number or a variable may have an index. Index of a variable (or a constant) is a value that is raised to the power of the variable. The indices are also known as powers or exponents. It shows the number of times a given number has to be multiplied. It is represented in the form:
a^{m }= a × a × a ×……× a (m times) |
Here, a is the base and m is the index.
The index says that a particular number (or base) is to be multiplied by itself, the number of times equal to the index raised to it. It is a compressed method of writing big numbers and calculations.
Example: 2^{3} = 2 × 2 × 2 = 8
In the example, 2 is the base and 3 is the index.
Laws of Indices
There are some fundamental rules or laws of indices which are necessary to understand before we start dealing with indices. These laws are used while performing algebraic operations on indices and while solving the algebraic expressions, including it.
Rule 1: If a constant or variable has index as ‘0’, then the result will be equal to one, regardless of any base value.
a^{0} = 1 |
Example: 5^{0} = 1, 12^{0} = 1, y^{0}= 1
Rule 2: If the index is a negative value, then it can be shown as the reciprocal of the positive index raised to the same variable.
a^{-p} = 1/a^{p} |
Example: 5^{-1} = ⅕, 8^{-3}=1/8^{3}
Rule 3: To multiply two variables with the same base, we need to add its powers and raise them to that base.
a^{p}.a^{q} = a^{p+q} |
Example: 5^{2}.5^{3} = 5^{2+3} = 5^{5}
Rule 4: To divide two variables with the same base, we need to subtract the power of denominator from the power of numerator and raise it to that base.
a^{p}/a^{q} = a^{p-q} |
Example: 10^{4}/10^{2} = 10^{4-2} = 10^{2}
Rule 5: When a variable with some index is again raised with different index, then both the indices are multiplied together raised to the power of the same base.
(a^{p})^{q} = a^{pq} |
Example: (8^{2})^{3} = 8^{2.3} = 8^{6}
Rule 6: When two variables with different bases, but same indices are multiplied together, we have to multiply its base and raise the same index to multiplied variables.
a^{p}.b^{p} = (ab)^{p} |
Example: 3^{2}.5^{2} = (3 x 5)^{2 }= 15^{2}
Rule 7: When two variables with different bases, but same indices are divided, we are required to divide the bases and raise the same index to it.
a^{p}/b^{p} = (a/b)^{p} |
Example: 3^{2}/5^{2} = (⅗)^{2}
Rule 8: An index in the form of a fraction can be represented as the radical form.
a^{p/q} = ^{q}√a^{p} |
Example: 6^{1/2} = √6
Indices Maths Problems
Q.1: Multiply x^{4}y^{3}z^{2} and xy^{5}z^{-1}
Solution: x^{4}y^{3}z^{2} and xy^{5}z^{-1}
= x^{4}.x .y^{3}.y^{5}.z^{2}.z^{-1}
= x^{4+1}.y^{3+5}.z^{2-1}
= x^{5}.y^{8}.z
Q.2: Solve a^{3}b^{2}/a^{2}b^{4}
Solution: a^{3}b^{2}/a^{2}b^{4}
= a^{3-2}b^{2-4}
= a^{1}b^{-2}
= a b^{-2}
= a/b^{2}
Q.3: Find the value of 27^{2/3}.
Solution: 27^{2/3}
= ^{3}√27^{2}
= 3^{2}
= 9