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.

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**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:

**a ^{n}**

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:**

2^{3 }

In this example, the base is 2, and the exponent is 3. The expression 2^{3 }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 | 8^{4}, Base = 8, Exponent = 4 |

5 × 5 × 5 × 5 × 5 × 5 × 5 | 5^{7}, Base = 5, Exponent = 7 |

3 × 3 × 3 | 3^{3}, 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 | 5^{0} |
1 | 1 |

One exponent | 7^{1} |
7 | 7 |

Exponent and power | 4^{3} |
4 × 4 × 4 | 64 |

Negative exponent | 3^{-3} |
1/3^{3} = 1/(3 × 3 × 3) |
1/27 |

Rational exponent | 16^{1/2} |
√16 | 4 |

Multiplication | 4^{2} × 4^{3} |
4^{(2 + 3)} = 4^{5} |
1024 |

Quotient | 8^{5}/ 8^{3} |
8^{(5 – 3)} = 8^{2} |
64 |

Power of exponent | (3^{2})^{2} |
3^{(2 × 2)} = 3^{4} |
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 (4^{2} × 4^{-5})/ 16^{-2}

**Solution:**

(4^{2} × 4^{-5})/ 16^{-2} = 4^{(2 – 5)} × 16^{2}

= 4^{-3} × (4^{2})^{2}

= 4^{-2} × 4^{4}

= 4^{(-2 + 4)}

= 4^{2}

= 16

**Example 2:** Simplify and write the answer in exponential form.

(3^{5} ÷ 3^{8} )^{5} × 3^{-5}

**Solution:**

(3^{5} ÷ 3^{8} )^{5} × 3^{-5}

= (3^{5-8})^{5} × 3^{-5}

= (3^{-3})^{5} × 3^{-5}

= 3^{(-15 – 5)}

=3^{-20}

Or

= 1/(3)^{20}