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What is an Exponent?

Exponentiation is a mathematical operation involving a base (a) and an exponent (n). Exponentiation corresponds to repeated multiplication of the base, n times.

Example: aⁿ = a × a × ... × a (n times)

Basic Exponent Laws and Rules

• When exponents with the same base are multiplied, the exponents are added: aⁿ × aᵐ = a^(n+m)
• When an exponent is negative, it's removed by reciprocating the base: a^(-n) = 1/aⁿ
• When exponents with the same base are divided, the exponents are subtracted: aᵐ / aⁿ = a^(m-n)
• When exponents are raised to another exponent, the exponents are multiplied: (aᵐ)ⁿ = a^(m×n)
• When multiplied bases are raised to an exponent, the exponent is distributed to both bases: (a × b)ⁿ = aⁿ × bⁿ
• When divided bases are raised to an exponent, the exponent is distributed to both bases: (a/b)ⁿ = aⁿ / bⁿ
• When an exponent is 1, the base remains the same: a¹ = a
• When an exponent is 0, the result is always 1: a⁰ = 1 (usually)
• When an exponent is a fraction with the numerator as 1, it's the nth root of the base: a^(1/n) = √a

Exponents with Negative Bases

Exponents with negative bases follow similar rules as positive bases. Exponents with negative bases raised to positive integers have the same magnitude as their positive counterparts but vary in sign.