Numbers and Operations · 4 min read
Exponents, Radicals, and Scientific Notation · Number Theory
Integer Exponents
| Rule | Statement | Example |
|---|---|---|
| Product | aᵐ · aⁿ = a^(m + n) | x³ · x² = x⁵ |
| Quotient | aᵐ ÷ aⁿ = a^(m − n) | 2⁷ ÷ 2⁴ = 2³ = 8 |
| Power of a power | (aᵐ)ⁿ = a^(mn) | (x²)³ = x⁶ |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2x²)³ = 8x⁶ |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (2/3)² = 4/9 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a^(−n) = 1/aⁿ and (a/b)^(−n) = (b/a)ⁿ | 2^(−3) = 1/8; (2/5)^(−2) = 25/4 |
To simplify an expression with several bases, rewrite every base as a power of the same prime. For example, (4^(−1) · 8²) ÷ 2^(−2) = (2^(−2) · 2⁶) ÷ 2^(−2) = 2⁶ = 64.
Square Roots and Radicals
Simplify a square root by removing perfect-square factors: √50 = √(25 · 2) = 5√2. Only like radicals (same radicand) can be added: 3√2 + 5√2 = 8√2, but √2 + √3 does not simplify. The product rule √a · √b = √(ab) holds for a, b ≥ 0, so √3 · √12 = √36 = 6.
Scientific Notation
A number in scientific notation has the form a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer: 5,400,000 = 5.4 × 10⁶ and 0.00032 = 3.2 × 10^(−4).
Divisibility Rules
| Divisor | Test |
|---|---|
| 2 | Last digit is even |
| 3 | Sum of digits is divisible by 3 |
| 4 | Number formed by the last two digits is divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Number formed by the last three digits is divisible by 8 |
| 9 | Sum of digits is divisible by 9 |
| 10 | Last digit is 0 |
Primes, Composites, and Prime Factorization
A prime number is a natural number greater than 1 with exactly two factors, 1 and itself (2, 3, 5, 7, 11, 13, …); 2 is the only even prime. A composite number has more than two factors. The number 1 is neither prime nor composite. By the Fundamental Theorem of Arithmetic, every integer greater than 1 has exactly one prime factorization: 60 = 2² · 3 · 5.
To test whether n is prime, you need only check primes up to √n: if n had a factor larger than √n, it would also have a matching factor smaller than √n. For 323, check 2, 3, 5, 7, 11, 13, and 17 (since √323 ≈ 18.0); 323 = 17 × 19, so it is composite.
GCF and LCM
The greatest common factor (GCF) is the product of the shared primes, each to the lowest power; the least common multiple (LCM) is the product of all primes that appear, each to the highest power. For 24 = 2³ · 3 and 36 = 2² · 3²: GCF = 2² · 3 = 12 and LCM = 2³ · 3² = 72. For any two positive integers, GCF(a, b) × LCM(a, b) = a × b.
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