1.2: Exponents and Scientific Notation

Product rule of exponents

If multiplying powers with the same base, add the exponents: a^m · a^n = a^(m+n). Example: x^3 · x^5 = x^8.

Quotient rule of exponents

If dividing powers with the same base, subtract the exponents: a^m / a^n = a^(m−n). Example: x^7 / x^3 = x^4.

Power rule of exponents

When raising a power to another power, multiply the exponents: (a^m)^n = a^(mn). Example: (x^3)^4 = x^12.

Zero exponent rule

Any nonzero number raised to the zero power equals 1: a^0 = 1. Example: 7^0 = 1.

Negative exponent rule

A negative exponent means take the reciprocal: a^(−n) = 1/a^n. Example: x^−3 = 1/x^3.

Power of a product

When a product is raised to a power, apply the exponent to every factor: (ab)^n = a^n b^n. Example: (2x)^3 = 8x^3.

Power of a quotient

When a quotient is raised to a power, apply the exponent to the numerator and denominator: (a/b)^n = a^n/b^n. Example: (x/2)^3 = x^3/8.

Simplifying exponential expressions

Rewrite expressions using exponent rules, then combine like bases and evaluate numerical powers when appropriate. Example: x^3 · x^2 / x^4 = x.

Scientific notation

Write a number as a × 10^n, where 1 ≤ |a| < 10 and n is an integer. For large numbers, n is positive; for small decimals, n is negative. Example: 0.00045 = 4.5 × 10^−4.