In mathematics, an Index is a power or exponent which is raised to a variable or number. Want to master solving large number index calculations in algebra? Then, go with these laws of indices article and memorize all the rules of indices with examples. In algebra, the index is dealing with regard to numbers. Let’s dive into further modules and learn about the laws/rules of the indices with formulas and worked-out examples.
Do Refer:
- Exponents and Indices
- Number Formed by Any Power
- Expansion of Powers of Binomials and Trinomials
- Application Problems on Expansion of Powers of Binomials and Trinomials
- Worksheet on Powers of Literal Numbers
Definition of Index
A variable or a number may have an index which is a value that raises the power of a number or variable. Powers or exponents are also called Indices. In short, indices show the number of times a given number has to be multiplied. Also, it is called to be a compressed method of writing large numbers and calculations. The representation of the index is in the form:
an = a x a x a x a x……..x a (n times)
Where a is the base and n is the index.
Laws of Indices – Rules & its Explanation
We have so many primary rules or laws of indices that are important to grasp before we start knowing about indices. These index laws are utilized while solving algebraic expressions and performing the algebraic operations on indices.
Rule 1: In case a constant or variable has index ‘0’, then the result will be equal to one, irrespective of any base value.
a0 = 1
Rule 2: If the negative value is represented as an index, then it should be shown as the reciprocal of the positive index of the same variable.
a-p = 1/ap
Rule 3: Whenever we have two variables with the same base and to multiply them, we have to add their powers or indices raised to the respective base.
ap.aq = ap+q
Rule 4: If you want to divide the two variables with the same base, we have to subtract the power of the denominator from the numerator power and raise it to the base.
ap/aq = ap-q
Rule 5: When a variable with some index is again raised with another index, then we have to multiply both indices together raised to the power of the same base.
(ap)q = apq
Rule 6: When we have two variables with the same indices, but different bases are multiplied together. First, we have to multiply the bases and raise the same index to it.
ap.bp = (ab)p
Rule 7: If two variables with various bases, but the same indices are divided, we have to divide the bases and raise the same index to it.
ap/bp = (a/b)p
Rule 8: An index in the form of a fraction can be denoted as the radical form.
ap/q = q√ap
List of Laws of Indices Formula
The list of important index laws is given here in the shareable image format, kindly make use of it for quick reference before exams and memorize the laws frequently.
Indices Examples | Questions on Laws of Indices
Example 1:
Find the numerical value for each of the following (not containing exponents):
(i) 80
(ii)5-10
(iii)272/3
Solution:
(i)80 = 1 [by using the law of indices].
(ii)5-10 = 1/510 = 1/5x5x5x5x5x5x5x5x5x5 = 1/9765625.
(iii)272/3 = 3√272 = 32 = 9.
Example 2:
Write the following expressions more concisely by using an index.
(i) b x b x b x b (ii) (x/y) x (x/y) (iii) ac x ac x ac x ac x ac
Solution:
(i) b x b x b x b = b4
(ii) (x/y) x (x/y) = (x/y)2
(iii) ac x ac x ac x ac x ac = (ac)5
Example 3:
Multiply x2y4z3 and xy3z-1
Solution:
Given that x2y4z3 and xy3z-1
= x2.x3 .y4.y3.z3.z-1
= x2+3.y4+3.z3-1
= x5.y7.z2
FAQs on Index Laws
1. What are the laws of indices with examples?
Laws of Indices are the rules for simplifying expressions including powers of the same base number. For instance, (−2)³ = −8 and (−2)4 = 16, so (−x)7 = −x7 and (−x)8 = -x8.
2. How many index laws are there mainly?
Majorly, there are three laws of indices. They are as such:
- am x an= am+n
- \(\frac { am }{ an } \) = am-n
- (am)n = a mn
3. How do you explain an index?
A number that raises to power is called an index number. The power, also known as the index, states you know how many times you have to multiply the number by itself. As an example, 5 means that you have to multiply 2 by itself five times = 2×2×2×2×2 = 32.