Conjugate (square roots)

In mathematics, particularly when dealing with algebraic expressions involving square roots, the conjugate of a binomial expression of the form a + b√c (where a and b are rational numbers and c is a non-negative rational number) is a - b√c. Similarly, the conjugate of a - b√c is a + b√c.

The key property of conjugates is that when multiplied together, they eliminate the square root term. Specifically, using the difference of squares factorization:

( a + b√c ) ( a - b√c ) = a² - (b√c)² = a² - b²c

Since a, b, and c are rational numbers, a² - b²c is also a rational number, free from any square roots.

This property is particularly useful in:

• Rationalizing the denominator: If a fraction has an irrational denominator containing a square root, multiplying both the numerator and denominator by the conjugate of the denominator will eliminate the square root from the denominator, making it a rational number. This process simplifies calculations and is often required for expressing mathematical answers in a standard form.
• Solving equations: Conjugates can be used to manipulate equations involving square roots, often leading to simpler forms that are easier to solve.
• Simplifying expressions: Multiplying an expression by its conjugate can sometimes lead to a simpler equivalent expression.

The concept extends beyond simple square roots. The conjugate can be generalized to other irrational expressions, although the specific form of the conjugate depends on the expression. For example, with cube roots or more complex expressions, the construction of the conjugate becomes more involved.