Introduction
Indices or Exponents represent how many times a number is multiplied by itself. It is a mathematical shorthand to write repeated multiplication. For example, instead of writing 5 × 5 × 5, we write 5 raised to the power of 3 (53).
In the expression 53, 5 is the Base and 3 is the Index or Exponent.
Explanation Step by Step
When a number is multiplied several times, the number being multiplied is called the Base. The number of times it is multiplied is called the Index or Power.
Sub-topics
Operations on numbers - Indices
Indices follow specific rules called Laws of Indices. These allow us to simplify complex calculations involving repeated multiplication and division quickly.
Common Equation Basis
For any non-zero integers 'a' and 'b', and whole numbers 'm' and 'n':
- am × an = am+n
- am ÷ an = am-n
- (a × b)m = am × bm
- (am)n = am × n
Examples
Example 1: Multiplication Law
Tricky Example: Power of a Power
Is (23)2 the same as 2(32)?
Tricks and Shortcuts
- The Power of Zero: Any non-zero number raised to 0 is always 1. Example: 9990 = 1.
- Negative Base Rule: If the index is Even, the answer is positive. If the index is Odd, the answer is negative. Example: (-1)50 = 1, but (-1)51 = -1.
- Unit Digit Trick: If the base ends in 0, 1, 5, or 6, the unit digit of the answer will always be that same digit (0, 1, 5, or 6) no matter what the index is!
Common Mistakes
- Adding Bases: Thinking that 23 + 22 = 25. Wrong! Laws of indices only apply to multiplication and division, not addition.
- Multiplying Base by Index: Doing 32 = 6. Correct is 3 × 3 = 9.
Practice Questions
Easy Questions
- Find the value of 104.
- Simplify: 57 ÷ 55.
- Tricky Question: Find the value of (5 + 5 + 5)0.
Medium Questions
- Simplify: (32)3.
- Real-life Question: A digital storage folder doubles in size every day. If it starts at 22 MB, what is its size after 3 days?
- Tricky Question: Which is greater: 52 or 25?
Hard Questions
- Solve: (23 × 22) ÷ 24.
- Real-life Tricky Question: An investor's money triples every year. Express the money after 5 years in index form if he started with 1 Rupee.
- Tricky Question: Simplify: (-1)1000 × (-1)1001.
Revision Summary
Indices simplify repeated multiplication. Key rules: add powers for multiplication, subtract for division, and multiply for power of a power. Any number to the power of 0 is 1. Be careful with brackets and addition!