Introduction to Exponents

An exponent represents the number of times a number (the base) is multiplied by itself. For example, 2³ means 2 × 2 × 2 = 8.

Exponents are a fundamental concept in mathematics and are widely used in scientific notation, compound interest, and even in understanding exponential growth.

Laws of Exponents

Law	Formula	Description
Product of Powers	a^m × aⁿ = a^m+n	Multiply like bases by adding exponents.
Quotient of Powers	a^m ÷ aⁿ = a^m-n	Divide like bases by subtracting exponents.
Power of a Power	(a^m)ⁿ = a^m×n	Raise a power to a power by multiplying exponents.
Zero Exponent	a⁰ = 1	Any non-zero number raised to the power of zero equals 1.
Negative Exponent	a^-n = 1/aⁿ	Represents the reciprocal of the positive exponent.

Variable Expressions with Exponents

Multiplication of Powers: x³ × x² = x⁵
Division of Powers: x⁵ ÷ x³ = x²
Power of a Power: (x²)³ = x⁶
Zero Exponent: x⁰ = 1
Negative Exponent: x^-2 = 1/x²
Power of a Product: (xy)³ = x³y³
Power of a Quotient: (x/y)² = x²/y²

Examples

1. (x²)³ × x⁴ = x¹⁰

2. y⁵ / y² = y³

3. 16^3/4 = √[4]{16³} = 8

Understanding Rational Exponents

Rational exponents a^m/n mean the n-th root of a raised to the m-th power: a^m/n = √[n]{a^m}.

16^1/2 = 4
27^2/3 = 9
8^-2/3 = 1/4

Exponential Equations

Same base: 2^x = 2⁵ ⇒ x = 5
Logs: 3^x = 7 ⇒ x = ln(7)/ln(3)

Exponent Calculator

Enter an expression (use ^ for powers, sqrt() for roots), e.g. 2^3 + sqrt(49).

Exponent Quiz

Select the correct answers. Five random questions each load.

Prime Factors — Quick Quiz

What are the prime factors of 30?

Concept Deep Dive

Order of Operations with Exponents

In PEMDAS/BODMAS, exponentiation happens before multiplication/division. E.g., 3 ⋅ 2^3 = 3 ⋅ 8 = 24, not (3⋅2)^3.

Why the Zero Exponent is 1

Using the quotient law with same base: \[ \frac{a^n}{a^n}=a^{n-n}=a^0 \quad\text{but}\quad \frac{a^n}{a^n}=1 \Rightarrow a^0=1 \quad(a\ne 0). \]

Negative & Rational Exponents

Negative: \(a^{-n}=\frac{1}{a^n}\). Example: \(10^{-3}=\frac{1}{1000}\).
Rational: \(a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m\). Example: \(27^{2/3}=\sqrt[3]{27^2}=9\).

Common Pitfalls

\( (a+b)^n \neq a^n + b^n \) in general (only true for special cases).
\( \sqrt{a^2}=|a| \), not \(a\) (principal square root is non-negative).
\( a^{m/n} \) assumes \(a\ge 0\) for even \(n\) in real numbers.

Exponential vs Polynomial Growth visual

Exponential functions eventually outgrow any polynomial. The chart below shows \(y=2^x\) and \(y=x^3\) on a small interval; \(2^x\) accelerates rapidly.

Exponent Reference Tables

Handy powers for quick checks and mental math.

Base \\ Power	0	1	2	3	4	5	6	7	8	9	10

Squares & Cubes to Memorize

n	n²	n³

Worked Examples (Advanced)

1) Simplify a Mixed Expression

\[ \frac{(2x^{-3}y^{1/2})^2 \cdot x^5}{4x^{-1}y} \]

Show steps

\[ (2x^{-3}y^{1/2})^2 = 2^2 \cdot x^{-6} \cdot y^{1} = 4x^{-6}y \] So, \[ \frac{4x^{-6}y \cdot x^5}{4x^{-1}y} = \frac{4x^{-1}y}{4x^{-1}y}=1. \] Answer: 1

2) Solve an Exponential Equation (logs on both sides)

\[ 5^{2x-1}=3^{x+2} \]

Show steps

Take natural logs: \[ (2x-1)\ln 5 = (x+2)\ln 3 \Rightarrow x(2\ln5-\ln3)=\ln3+ \ln5 \Rightarrow x=\frac{\ln3+\ln5}{2\ln5-\ln3}. \] Exact: \(\displaystyle \frac{\ln(15)}{2\ln5-\ln3}\).

3) Rational Exponents with Domain

\[ \left( x^{\frac{3}{2}}\right)^{\frac{2}{3}} = x \quad (\text{with } x\ge 0) \]

Why the domain matters

\((x^{3/2})^{2/3}=x^{(3/2)\cdot(2/3)}=x^1\). For real numbers, \(x^{1/2}\) requires \(x\ge0\). Answer: \(x\) for \(x\ge 0\).

4) Geometric Series Connection

Sum \(S=1+2+2^2+\cdots+2^{n}=\dfrac{2^{n+1}-1}{2-1}=2^{n+1}-1\).

5) Compound Interest

\(A=P\left(1+\frac{r}{n}\right)^{nt}\). If \(P=5000\), \(r=8\%\), \(n=12\), \(t=5\):

Compute

\[ A=5000\left(1+\frac{0.08}{12}\right)^{12\cdot5} \approx 5000(1.006\overline{6})^{60}\approx 5000\cdot 1.48985\approx \mathbf{R7\,449.25}. \]

Exponents Course