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Index Questions and Answers - Concepts, Formulas, FAQs | Testbook
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In maths, the term index (or indices) means how many times a number or variable is multiplied by itself. For example, in 34, the index is 4, which means 3 is multiplied by itself 4 times. This topic is important because it helps us work with large numbers and expressions easily. In this article, you’ll find different questions on indices to help you practise and understand how to solve problems involving powers of numbers, variables, and special expressions.
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What does Index mean in Mathematics?
In mathematics, an index (also called an exponent or power) shows how many times a number is multiplied by itself. For example, in the expression 3^5, the index is 5. This means that 3 is multiplied by itself five times: 3 × 3 × 3 × 3 × 3. Index numbers are used to write large multiplications in a shorter and simpler form. They help make calculations easier and are commonly used in algebra and other branches of math. Understanding indices is important for solving many types of mathematical problems quickly and efficiently.
Also, read: Index
Laws of Indices
- a m × a n = a m+n
- a m /a n = a m-n
- a 0 = 1
- a -m = 1/a m
- a m × b m = (a × b) m
- (a m ) n = a mn
- a 1/2 = √a
Practice Problems on Indices
Example 1: Evaluate: 50 6 ÷ 50 4
Solution:
50 6 ÷ 50 4
Applying the formula a m /a n = a m-n , we get;
50 6 ÷ 50 4 = 50 6 / 50 4
= 50 (6-4)
= 50 2
= 50 × 50
= 2500
Hence, 50 6 ÷ 50 4 = 2500.
Example 2: If 2 (x-y) = 16 and 2 (x+y) = 256, determine the values of x and y.
Solution:
Given,
2 (x-y) = 16
2 (x-y) = 2 4
Since the bases are equal, equate the powers (indices).
Thus, x – y = 4….(i)
and
2 (x+y) = 256
2 (x+y) = 2 8
Again, equate the powers (indices) as the bases are equal.
Hence, x + y = 8….(ii)
Adding equations (i) and (ii), we get;
2x = 12
x = 6
Substituting x = 6 in equation (ii), we get;
6 + y = 8
y = 8 – 6 = 2
Therefore, x = 6 and y = 2.
Example 3: Simplify the given expression: (a 2 ) 0 × (a 1/3 ) 3.
Solution:
(a 2 ) 0 × (a 1/3 ) 3
We know that a 0 = 1 and (a m ) n = a mn .
So, (a 2 ) 0 × (a 1/3 ) 3 = a (2×0) × a (1/3 × 3)
= a 0 × a 1
= 1 × a
= a
Therefore, (a 2 ) 0 × (a 1/3 ) 3 = a.
Example 4: Write 16^(x + 2) in the form of 2^y and find the relation between x and y.
Solution:
We know that 16 = 2⁴.
So,
16^(x + 2) = (2⁴)^(x + 2)
Now apply the rule (a^m)^n = a^(m × n):
= 2^(4 × (x + 2))
= 2^(4x + 8)
This is in the form 2^y, so:
y = 4x + 8
Example 5: Evaluate the expression: 272^(2/3) × 343^(1/2)
Solution:
We know:
- 27 = 3³
- 343 = 7³
So,
272^(2/3) × 343^(1/2)
= (3³)^(2/3) × (7³)^(1/2)
= 3^(3 × 2/3) × 7^(3 × 1/2)
= 3² × 7^(3/2)
= 9 × √(7³)
= 9 × √343
So, the simplified expression is:
9 × √343
(Note: You can leave it in this form or approximate √343 ≈ 18.52, so:
9 × 18.52 ≈ 166.68, if needed.)
Additional Practice Problems on Indices
- Evaluate (15) 6.5 x (3) 1.5 ÷ (45) 0.5 = 3 m . What is the value of m?
- Simplify the expression: a 2 b −1 × (a 1 b 1 ) 3
- If (a/b) (x-2) = (b/a) (x-4) , then what is the value of x?
- Simplify: (3x) -3 /x 3
- Evaluate: (9) 0.09 × (9) 0.01 × (3) 0.1
FAQs For Index Questions and Answers
What is an index in mathematics?
In mathematics, an index is a number that tells us how many times a particular number is multiplied by itself.
What are the laws of indices?
The laws of indices include a^m × a^n = a^(m+n), a^m /a^n = a^(m-n), a^0 = 1, a^-m = 1/a^m, a^m × b^m = (a × b)^m, (a^m)^n = a^(mn), a^(1/2) = √a.
How to solve index problems?
Index problems can be solved by applying the laws of indices and simplifying the expressions step by step.
What is the difference between index and power?
The terms "index" and "power" are often used to mean the same thing. Both refer to the exponent in a mathematical expression like aⁿ, where "n" is called the index or power.
What is the index form of a number?
Writing a number using exponents is called index form. Example: 81 = 3⁴, so the index form is 3⁴.
Why is a⁰ = 1?
This comes from the rule aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰. Since any number divided by itself is 1, we get a⁰ = 1.
How do you simplify expressions with indices?
Use the laws of indices (like the product rule, quotient rule, and power of a power rule) to combine terms with the same base and simplify the expression.
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