What is an Exponent?

An exponent represents the number of times a base number is multiplied by itself. The expression bⁿ means b is multiplied by itself n times.

Special Cases

Zero Exponent

Any non-zero number raised to the power of 0 equals 1.

Negative Exponents

A negative exponent means the reciprocal of the base raised to the absolute value of the exponent.

Fractional Exponents

Fractional exponents represent roots. The denominator of the fraction is the root, and the numerator is the power.

Special Forms

Expression	Result	Notes
00	Undefined or 1	Mathematically indeterminate
0n, n > 0	0	Zero raised to a positive power is zero
0n, n < 0	Undefined	Involves division by zero
1n, any n	1	One raised to any power is one
(-b)n, n is even	bn	Result is positive
(-b)n, n is odd	-bn	Result has same sign as -b
(-b)n, n is not an integer	Complex number	Involves complex arithmetic

Laws of Exponents

Law	Formula	Example
Product Rule	bx × by = bx+y	23 × 24 = 27 = 128
Quotient Rule	bx ÷ by = bx-y	25 ÷ 22 = 23 = 8
Power Rule	(bx)y = bxy	(23)2 = 26 = 64
Negative Exponent	b-x = 1 / bx	3-2 = 1 / 32 = 1/9 ≈ 0.111
Zero Exponent	b0 = 1 (b ≠ 0)	70 = 1
Fractional Exponent	b1/n = n√b	161/2 = √16 = 4
Distributive Property	(a × b)n = an × bn	(2 × 3)2 = 62 = 36, and 22 × 32 = 4 × 9 = 36

Applications of Exponents

Scientific Notation

Large and small numbers are often written in scientific notation: a × 10ⁿ, where 1 ≤ a < 10.

Compound Interest

When money grows with compound interest, the formula is A = P(1 + r)^t, where:

• A = final amount
• P = principal (initial investment)
• r = interest rate (decimal)
• t = time periods

Growth and Decay

Many natural processes follow exponential growth or decay, with formulas like: