Conjugates are pairs of expressions that have the same coefficients but opposite signs. They are particularly important in the context of factoring special products and performing operations on radical expressions.
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Conjugates are used to factor special products, such as the difference of squares, by pairing the terms with opposite signs.
When multiplying radical expressions, conjugates are used to eliminate the square root symbol and simplify the expression.
Adding or subtracting radical expressions with the same radicand can be done by using conjugates to create a common denominator.
The product of a number and its conjugate is always a perfect square.
Conjugates are useful for rationalizing the denominator of a fraction containing a radical expression.
Review Questions
Explain how conjugates are used to factor the difference of squares.
To factor the difference of squares, $a^2 - b^2$, we can use the conjugates $(a + b)$ and $(a - b)$. The factorization would be: $a^2 - b^2 = (a + b)(a - b)$. This works because the product of the conjugates is always a perfect square, $a^2 - b^2 = (a + b)(a - b)$.
Describe the process of using conjugates to simplify the multiplication of radical expressions.
When multiplying radical expressions with the same radicand, such as $\sqrt{a} \cdot \sqrt{b}$, we can use conjugates to eliminate the square root symbol and simplify the expression. The process involves multiplying the first expression by its conjugate, $\sqrt{a}$, and the second expression by its conjugate, $\sqrt{b}$. This results in $a \cdot b$, which is a perfect square and can be simplified further.
Analyze how conjugates can be used to rationalize the denominator of a fraction containing a radical expression.
To rationalize the denominator of a fraction containing a radical expression, such as $\frac{1}{\sqrt{a}}$, we can multiply both the numerator and denominator by the conjugate of the denominator, which is $\sqrt{a}$. This eliminates the square root symbol in the denominator, resulting in $\frac{\sqrt{a}}{a}$, which is a rational expression. Rationalizing the denominator is important because it allows for easier manipulation and calculation of the fraction.