Simplify Calculator

The Greatest Common Divisor (GCD), also called the Greatest Common Factor (GCF), is the largest positive integer that divides two or more numbers without leaving a remainder. It is the foundation of fraction simplification: once you know the GCD of a numerator and denominator, you divide both by that number to produce the equivalent fraction in lowest terms. The challenge lies in finding the GCD efficiently, especially for large numbers where trial division becomes impractical.

The Euclidean algorithm, described by the Greek mathematician Euclid around 300 BC, is the oldest and most elegant solution to this problem. The algorithm exploits a simple but powerful observation: the GCD of two numbers a and b equals the GCD of b and the remainder when a is divided by b. This observation allows us to repeatedly replace the larger number with the smaller number and the smaller number with the remainder, shrinking the problem at each step until the remainder is zero — at which point the last non-zero remainder is the GCD.

As a concrete example, consider finding GCF(48, 18). We compute 48 ÷ 18 = 2 remainder 12, so GCF(48, 18) = GCF(18, 12). Next, 18 ÷ 12 = 1 remainder 6, so GCF(18, 12) = GCF(12, 6). Finally, 12 ÷ 6 = 2 remainder 0, so GCF(12, 6) = 6. Therefore GCF(48, 18) = 6, and the fraction 48/18 simplifies to 8/3. The algorithm required only three steps — far fewer than checking every potential divisor by brute force.

The Euclidean algorithm's efficiency comes from the fact that the remainder shrinks rapidly at each step. It can be proven that after every two steps, the larger number shrinks by at least half, meaning the algorithm always finishes in a number of steps proportional to the number of digits in the input — making it extremely fast even for hundred-digit numbers. This property makes it the algorithm of choice in computational number theory, cryptography (RSA key generation), and symbolic math systems.

Beyond fractions, GCD appears in rational number arithmetic, modular arithmetic, and solving linear Diophantine equations of the form ax + by = c. By Bezout's identity, the GCD of a and b can always be expressed as a linear combination of a and b with integer coefficients — a fact that the extended Euclidean algorithm computes alongside the GCD itself, enabling modular inverses used in cryptography and error-correcting codes.

Simplifying radical expressions means rewriting them in a form where no perfect square (or perfect cube, for cube roots) remains under the radical sign. The fundamental property that makes this possible is the product rule for radicals: √(a × b) = √a × √b. By factoring the radicand into a perfect square and a remainder, we can extract the perfect square as an integer coefficient outside the radical.

The most direct method is the perfect square factor approach. To simplify √72, look for the largest perfect square that divides 72. The perfect squares are 4, 9, 16, 25, 36, 49, 64 — and 36 divides 72 evenly: 72 = 36 × 2. Therefore √72 = √(36 × 2) = √36 × √2 = 6√2. Notice that working with the largest perfect square factor immediately gives the fully simplified result. If you had used 4 instead (72 = 4 × 18), you would get √72 = 2√18, and then need to simplify √18 = √(9 × 2) = 3√2 further, arriving at 2 × 3√2 = 6√2 — the same answer but in two steps.

The prime factorization approach provides a systematic alternative: factor the radicand completely into primes, then group factors in pairs. For √72: 72 = 2 × 2 × 2 × 3 × 3. Each pair of identical primes contributes one factor outside the radical: the pair of 2s gives 2, the pair of 3s gives 3, and the lone remaining 2 stays under the radical. Result: 2 × 3 × √2 = 6√2. This method handles any radicand systematically and extends naturally to cube roots (group in triples) and higher-order roots.

For cube root simplification, √[3]54: 54 = 2 × 3 × 3 × 3. The triple of 3s extracts as 3 outside the cube root, leaving √[3]2. So √[3]54 = 3√[3]2. Rationalizing denominators is another important technique: when a radical appears in the denominator, multiply numerator and denominator by the same radical to clear it. For 1/√2, multiply by √2/√2 to get √2/2 — a standard simplified form that avoids division by an irrational number.

Nested radicals like √(5 + 2√6) can sometimes be simplified using the identity √(a + b + 2√(ab)) = √a + √b. Here, 5 + 2√6 = 3 + 2 + 2√(3×2) = (√3 + √2)², so √(5 + 2√6) = √3 + √2. Recognizing such structures requires pattern matching and algebraic fluency, but the underlying tools — product rule, prime factorization, and algebraic identities — are the same as in every other radical simplification.