Simplifying mathematical expressions
Introduction
On this page, there are examples of simplifying mathematical expressions by moving square roots from the denominator to the numerator.
In the formulas, a, b and c are real numbers.
Involving the golden ratio
The golden ratio (phi): φ = (1 + √5) / 2 ≈ 1.618
| 1 | / | φ | = | φ − 1 | |||
| 1 | / | (φ + 1) | = | 2 − φ | |||
| 1 | / | (φ − 1) | = | φ | |||
| 1 | / | (φ + a) | = | (a + 1 − φ) | / | (a(a + 1) − 1) | |
| 1 | / | (φ + 1 / a) | = | a(a(φ − 1) − 1) | / | (a(a − 1) − 1) | |
| 1 | / | (φ + a / b) | = | b(a + b(1 − φ)) | / | (a2 + ab − b2) | |
| 1 | / | (aφ) | = | (φ − 1) | / | a | |
| 1 | / | (aφ + 1) | = | (a(φ − 1) − 1) | / | (a(a − 1) − 1) | |
| 1 | / | (aφ − 1) | = | (a(φ − 1) + 1) | / | (a(a + 1) − 1) | |
| 1 | / | (aφ + b) | = | (a(φ − 1) − b) | / | (a2 − ab − b2) | |
| 1 | / | (aφ + 1 / b) | = | b(ab(φ − 1) − 1) | / | (ab(ab − 1) − 1) | |
| 1 | / | (aφ + b / c) | = | c(ac(φ − 1) − b) | / | (a2c2 − b2 − abc) | |
Example: 1 / (3φ + 5/7) = 7(21φ − 26) / 311
Logarithms
| loga(a) | = | 1 | |
| loga(bc) | = | loga(b) + loga(c) | |
| loga(ab) | = | 1 + loga(b) | |
| loga(b / c) | = | loga(b) − loga(c) | |
| loga(a / b) | = | 1 − loga(b) | |
| loga(bc) | = | loga(b) × c | |
| loga(ab) | = | b | |
| loga(abcd) | = | b + loga(c) × d | |
| loga(√b) | = | loga(b) / 2 | |
| loga(∛b) | = | loga(b) / 3 | |
| loga(b) / loga(c) | = | logc(b) | |
| logab(a) | = | 1 − logab(b) | |
| logab(ac) | = | 1 − logab(b) + logab(c) |
Other expressions
| 1 | / | √a | = | √a | / | a | |
| 1 | / | (a + √b) | = | (a − √b) | / | (a2 − b) | |
| 1 | / | (a − √b) | = | (a + √b) | / | (a2 − b) | |
| 1 | / | (1 / a + √b) | = | a(1 − a√b) | / | (1 − a2b) | |
| 1 | / | (1 / a − √b) | = | a(1 + a√b) | / | (1 − a2b) | |
| 1 | / | (a / b + √c) | = | b(a − b√c) | / | (a2 − b2c) | |
| 1 | / | (a / b − √c) | = | b(a + b√c) | / | (a2 − b2c) | |
| 1 | / | (√a + √b) | = | (√a − √b) | / | (a − b) | |
| 1 | / | (√a − √b) | = | (√a + √b) | / | (a − b) |
Example: 1 / (3 + √7) = (3 − √7) / 2