conjugates
In mathematics, conjugates refer to two expressions obtained by changing the sign between the terms
In mathematics, conjugates refer to two expressions obtained by changing the sign between the terms. More specifically, if we have an expression of the form a + b, then its conjugate is a – b. Similarly, if we have an expression of the form a – b, then its conjugate is a + b.
Conjugates are commonly used in algebra to simplify and manipulate expressions. Their main purpose is to eliminate radicals in the denominator of a fraction. This technique is known as rationalizing the denominator.
For example, consider the fraction (3 + √2)/(2 – √3). To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2 – √3 is 2 + √3. Thus, multiplying the fraction by the conjugate, we get:
[(3 + √2)(2 + √3)] / [(2 – √3)(2 + √3)].
Expanding the numerator and denominator, we have:
(6 + 3√2 + 2√3 + √6) / (4 – 3).
Simplifying further, we obtain:
(9 + 3√2 + 2√3 + √6) / 1.
So, the expression (3 + √2)/(2 – √3) has been rationalized by multiplying it by its conjugate. Note that the denominator no longer contains a radical.
Conjugates can also be used in solving equations, particularly when dealing with complex numbers. For example, the conjugate of a complex number a + bi is a – bi. This property is used to find the roots of complex numbers with non-zero imaginary parts.
More Answers:
Understanding Radical Expressions | Definition, Examples, and Simplification Techniques in MathSimplifying Fractions to the Simplest Form | Understanding and Methodology
How to Rationalize the Denominator | A Step-by-Step Guide with Examples