10.1 - Introduction to Exponents and Powers

In this section, we will introduce the concept of exponents and powers, which are fundamental in understanding how numbers can be expressed in a compact form. The concept of exponents and powers helps in simplifying large numbers and expressing repeated multiplication of a number in a more manageable way.

An exponent (or power) is a shorthand notation to represent repeated multiplication of the same number. For example, instead of writing the number 2 multiplied by itself 5 times, we can simply write it as $2^5$. Here, 2 is the base, and 5 is the exponent (or power).

Exponent and Base

The general form of an expression involving an exponent is:

Base $^{\text{Exponent}}$.

For example, in the expression $3^4$, 3 is the base, and 4 is the exponent. This means that 3 is multiplied by itself 4 times, i.e.,

$3^4 = 3 \times 3 \times 3 \times 3 = 81$.

Types of Exponents

• Positive Exponent: When the exponent is a positive integer, it indicates repeated multiplication. For example, $2^3 = 2 \times 2 \times 2 = 8$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $5^0 = 1$.
• Negative Exponent: A negative exponent indicates the reciprocal of the number raised to the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.
• Fractional Exponent: An exponent in the form of a fraction indicates a root of a number. For example, $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$.

Law of Exponents

There are several important laws of exponents that help simplify expressions involving exponents:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$.
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.
• Power of a Power Rule: When raising a power to another power, multiply the exponents: $(a^m)^n = a^{m \times n}$.
• Power of a Product Rule: When raising a product to a power, distribute the exponent to each factor: $(a \times b)^m = a^m \times b^m$.
• Power of a Quotient Rule: When raising a quotient to a power, distribute the exponent to both the numerator and the denominator: $\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$.

Examples

• Example 1: Simplify $2^3 \times 2^4$:

  Using the Product of Powers Rule: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$.

• Example 2: Simplify $\frac{3^5}{3^2}$:

  Using the Quotient of Powers Rule: $\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$.

• Example 3: Simplify $(2^3)^2$:

  Using the Power of a Power Rule: $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$.