Exponents

Exponents: What Are They?

Sometimes, when we need to multiply a number many times, such as multiplying 2 by itself 100 times, writing each multiplication individually and performing the operation would not be practical. Instead, we use exponential notation to express such multiplications in a shorter form. Exponential expressions are shorthand for multiplying the same number repeatedly.

    \[a^n = a \times a \times a \times \dots \times a \quad (\text{n times})\]

Additionally, the concept of a factorial works in a similar way. To represent the product of multiple numbers, we use factorial notation. For example, the product of all integers from 1 to (n) is written as (n!) (factorial). This is expressed as:

    \[n! = 1 \times 2 \times 3 \times \dots \times n\]

This allows us to represent a large number of multiplications in a simplified form.

Exponential expressions and factorials are essential tools for efficiently writing and understanding mathematical operations.

Exponents provide a concise way to express repeated multiplication. A base number is multiplied by itself a specific number of times.

Example:

    \[5 \cdot 5 \cdot 5 \cdot 5 = 5^4\]


General Information About Exponents

Positive Exponents

A positive exponent indicates how many times the base is multiplied by itself.

    \[a^n = a \cdot a \cdot a \cdot \cdots \quad \text{(n times)}\]


Example:

    \[3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81\]

Negative Exponents

A negative exponent represents the reciprocal of the base raised to the corresponding positive power.

    \[a^{-n} = \frac{1}{a^n}, \quad \text{where } a \neq 0\]


Example:

    \[2^{-3} = \frac{1}{2^3} = \frac{1}{8}, \quad 5^{-2} = \frac{1}{5^2} = \frac{1}{25}\]

Zero Exponents

Any nonzero number raised to the power of zero equals 1:

    \[a^0 = 1, \quad \text{where } a \neq 0\]


Examples:

    \[3^0 = 1, \quad (-7)^0 = 1\]

Importance of Parentheses

Parentheses are critical when working with negative bases, as they determine how the exponent is applied:

    \[(-a)^n: \text{The exponent affects the entire negative base.}\]


    \[-a^n: \text{The exponent applies only to the base, not the negative sign.}\]


Examples:

    \[(-3)^2 = 9, \quad -3^2 = -(3^2) = -9\]


    \[(-3)^3 = -27, \quad -3^3 = -(3^3) = -27\]

Exponents represent how many times a number (the base) is multiplied by itself. For example:

    \[a^n = \underbrace{a \cdot a \cdot a \cdots}_{n \text{ times}}\]

Exponential Numbers Operations

1. Addition and Subtraction

Exponents can only be added or subtracted if the bases and exponents are the same:

    \[a^n + a^n = 2a^n\]

Example:

    \[3^2 + 3^2 = 2 \cdot 3^2 = 18\]

2. Multiplication

When multiplying exponents with the same base,add the powers:

    \[a^m \cdot a^n = a^{m+n}\]

Example:

    \[2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128\]

If the exponents are the same but the bases are different,multiply the bases:

    \[a^n \cdot b^n = (a \cdot b)^n\]

Example:

    \[2^3 \cdot 3^3 = (2 \cdot 3)^3 = 6^3 = 216\]

3. Division

When dividing exponents with the same base,subtract the powers:

    \[\frac{a^m}{a^n} = a^{m-n}\]

Example:

    \[\frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25\]

If the exponents are the same but the bases are different,divide the bases:

    \[\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n\]

Example:

    \[\frac{8^2}{4^2} = \left(\frac{8}{4}\right)^2 = 2^2 = 4\]

4. Power of a Power

When raising an exponent to another power,multiply the exponents:

    \[\left(a^m\right)^n = a^{m \cdot n}\]

Example:

    \[\left(3^2\right)^3 = 3^{2 \cdot 3} = 3^6 = 729\]

15 Jan, 2025