Exponents (Powers)

What this chapter covers

• Powers and exponents
• Laws of exponents
• The zero exponent
• Negative exponents
• Laws of exponents with integral powers
• Standard form of large and small numbers

Key points

• A power such as aⁿ has a base a and an exponent n; for a positive whole number n it means n factors of a multiplied together.
• A negative base gives a positive value for an even exponent and a negative value for an odd exponent; (-3)² = 9 but -3² = -9.
• The laws of exponents: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m-n, (a^m)ⁿ = a^mn, (ab)^m = a^m b^m, ((a)/(b))^m = (a^m)/(b^m).
• Any non-zero number to the power zero equals 1: a⁰ = 1.
• A negative exponent means a reciprocal: a^-m = (1)/(a^m) and ((a)/(b))^-m = ((b)/(a))^m.
• All the laws hold for every integer exponent when the bases are non-zero.
• To simplify, write each base as a power of a prime, collect exponents with the laws, then remove negative exponents.
• If a^x = a^y and a is not 0, 1 or -1, then x = y.
• Standard form is k × 10ⁿ with 1 ≤ k < 10 and n an integer: positive n for large numbers, negative n for numbers less than 1.

Questions answered