Exponents, a fundamental concept in mathematics, introduce a concise way of expressing repeated multiplication. In an expression like a^n , ‘a’ is the base, and ‘n’ is the exponent, indicating how many times ‘a’ should be multiplied by itself.
Delving deeper, a^n breaks down into two components: the base (‘a’) and the exponent (‘n’). The base is the number being multiplied, while the exponent denotes the number of times it is multiplied.
Table of Contents
Exponent Meaning
The term “exponent” refers to a mathematical notation used to express the repeated multiplication of a number by itself. It involves two key components: the base and the exponent.
Breaking It Down:
- Base:
- The base is the number that undergoes repeated multiplication.
- In the expression a^n, ‘a’ is the base.
- Exponent:
- The exponent is the small, raised number that indicates how many times the base is multiplied by itself.
- In a^n, ‘n’ is the exponent.
Exponent Symbol
The symbol used to denote an exponent in mathematics is the caret (^). The exponent is a small, raised number placed to the right and above the base number. Here’s how it looks in notation:
an
In this expression, ‘a’ is the base, and ‘n’ is the exponent. The caret symbol indicates that the base ‘a’ is raised to the power of ‘n’.
Example:
23
In this example, the base is 2, and the exponent is 3. The expression 23 represents the cube of 2, which is equal to 2 multiplied by itself three times (2 * 2 * 2), resulting in 8.
| Expression | Exponent representation |
| 8 × 8 × 8 × 8 | 84, Base = 8, Exponent = 4 |
| 5 × 5 × 5 × 5 × 5 × 5 × 5 | 57, Base = 5, Exponent = 7 |
| 3 × 3 × 3 | 33, Base = 3, Exponent = 3 |
The caret symbol is widely used in mathematical notation to express powers or exponents and is a fundamental element in algebraic expressions and equations.
Exponent Laws
The laws of exponents, also known as exponent rules, govern the manipulation and simplification of expressions involving exponents. These rules are fundamental in algebra and provide a systematic way to perform operations with exponential terms. Let’s delve into each law:
Exponent Table
| Type of Exponent | Expression | Expansion | Simplified value |
| Zero exponent | 50 | 1 | 1 |
| One exponent | 71 | 7 | 7 |
| Exponent and power | 43 | 4 × 4 × 4 | 64 |
| Negative exponent | 3-3 | 1/33 = 1/(3 × 3 × 3) | 1/27 |
| Rational exponent | 161/2 | √16 | 4 |
| Multiplication | 42 × 43 | 4(2 + 3) = 45 | 1024 |
| Quotient | 85/ 83 | 8(5 – 3) = 82 | 64 |
| Power of exponent | (32)2 | 3(2 × 2) = 34 | 81 |
Key Points:
- Repetition: Exponents embody the concept of repeated operations, simplifying expressions.
- Compact Notation: They offer a concise way to express large or repetitive multiplications.
- Versatility: Exponents extend beyond integers to include fractions, decimals, and even variables.
Solved Examples
Example 1: Simplify (42 × 4-5)/ 16-2
Solution:
(42 × 4-5)/ 16-2 = 4(2 – 5) × 162
= 4-3 × (42)2
= 4-2 × 44
= 4(-2 + 4)
= 42
= 16
Example 2: Simplify and write the answer in exponential form.
(35 ÷ 38 )5 × 3-5
Solution:
(35 ÷ 38 )5 × 3-5
= (35-8)5 × 3-5
= (3-3)5 × 3-5
= 3(-15 – 5)
=3-20
Or
= 1/(3)20
