CHAPTER 10 - EXPONENTS AND POWERS

An exponent refers to the number of times a number is multiplied by itself. For example:

\[ 5^4 = 5 \times 5 \times 5 \times 5 = 625 \]

Here, 5 is the base and 4 is the exponent or power.

Exponents help us:

• Write very large numbers (like the distance between stars) concisely
• Write very small numbers (like the size of an atom) conveniently
• Simplify complex calculations
• Express repeated multiplication efficiently

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

\[ a^m \times a^n = a^{m+n} \]

Example: \(2^3 \times 2^4 = 2^{3+4} = 2^7 = 128\)

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

\[ a^m \div a^n = a^{m-n} \]

Example: \(5^6 \div 5^2 = 5^{6-2} = 5^4 = 625\)

When raising a power to another power, multiply the exponents:

\[ (a^m)^n = a^{m \times n} \]

Example: \((3^2)^4 = 3^{2 \times 4} = 3^8 = 6561\)

When raising a product to a power, each factor gets raised to that power:

\[ a^m \times b^m = (ab)^m \]

Example: \(2^4 \times 3^4 = (2 \times 3)^4 = 6^4 = 1296\)

Any non-zero number raised to the power of zero equals 1:

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

Example: \(7^0 = 1\), \((-5)^0 = 1\), \((0.25)^0 = 1\)

When raising a quotient to a power, both numerator and denominator get raised to that power:

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

Example: \(\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}\)

A number with a negative exponent equals its reciprocal with a positive exponent:

\[ a^{-n} = \frac{1}{a^n} \]

Example: \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\), \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)

Scientific notation is a way to express very large or very small numbers using powers of 10:

\[ N \times 10^n \]

Where \(1 \leq N < 10\) and \(n\) is an integer.

Examples:

• \(300,000,000 = 3 \times 10^8\)
• \(0.00000045 = 4.5 \times 10^{-7}\)
• \(602,200,000,000,000,000,000,000 = 6.022 \times 10^{23}\) (Avogadro's number)

Exponents are used in many real-world applications:

• Computer Science: Measuring data storage (kilobytes, megabytes, gigabytes)
• Physics: Expressing very large distances (light-years) and very small measurements (atomic sizes)
• Finance: Calculating compound interest
• Biology: Modeling population growth
• Chemistry: Expressing concentrations (pH scale)

In this chapter, you will learn:

• How to read and write numbers using exponents
• The laws of exponents and how to apply them
• How to simplify expressions with exponents
• How to convert between standard form and scientific notation
• How to solve problems using exponents

This chapter includes exercises to help you practice:

• Expressing numbers in exponential form
• Applying the laws of exponents
• Converting between standard and scientific notation
• Comparing numbers with exponents
• Solving word problems using exponents