Problem 22 - Algebra 2

Question

Multiply each pair of conjugates. $$ (\sqrt{3}+\sqrt{5})(\sqrt{3}-\sqrt{5}) $$

Step-by-Step Solution

Answer

The product of the given pair of conjugates is -2.

Step 1: Identify the Conjugates

In this case, \(\sqrt{3} + \sqrt{5}\) and \(\sqrt{3} - \sqrt{5}\) are conjugates, meaning \(a = \sqrt{3}\) and \(b = \sqrt{5}\).

Step 2: Apply the Difference of Squares Formula

According to the difference of squares formula, \( (a+b)(a-b) = a^2 - b^2 \). Our a and b are \(\sqrt{3}\) and \(\sqrt{5}\). Therefore, multiplying out our formula we get: \(\sqrt{3}^2 - \sqrt{5}^2\)

Step 3: Simplify the Result

Simplify both the squares \(\sqrt{3}^2\) and \(\sqrt{5}^2\), resulting in 3 - 5.

Step 4: Final Solution

Finalize the calculation by resolving the subtraction, this results in -2.

Key Concepts

Conjugates

Conjugates are pairs of expressions that look alike but have opposite signs between their terms. In mathematical terms, if you have an expression of the form \(a + b\), its conjugate is \(a - b\). The beauty of conjugates lies in their ability to simplify expressions, especially when involving square roots or complex numbers.

By using conjugates, you can leverage the "Difference of Squares" property. This property states:

• \((a+b)(a-b) = a^2 - b^2\)

This effectively eliminates the middle term, simplifying the expression to just the difference of the squares of the terms involved. This is particularly useful because it allows us to remove the radicals (like square roots) when multiplying such expressions. In this exercise, the conjugates \(\sqrt{3} + \sqrt{5}\) and \(\sqrt{3} - \sqrt{5}\) simplify using this principle to yield a result free of square roots.

Square Roots

A square root is simply the inverse operation of squaring a number. If you square a number \(x\), you get \(x^2\). If \(y^2 = x\), then \(y\) is a square root of \(x\). Square roots are denoted by the radical sign \(\sqrt{}\). For instance, the square root of 9 is 3, since \(3^2 = 9\).

In the realm of radicals, the concept of simplifying square roots is key. When two radical expressions are conjugates, as in this exercise with \(\sqrt{3}\) and \(\sqrt{5}\), multiplying them using the difference of squares method allows us to eliminate the square roots through simplification. By squaring each term separately, you get:

• \(\sqrt{3}^2 = 3\)
• \(\sqrt{5}^2 = 5\)

resulting in a cleaner, straightforward expression without radicals.

Simplifying Expressions

Simplifying expressions is a fundamental part of algebra that involves reducing expressions to their simplest form. This often means removing square roots, combining like terms, or using algebraic identities. Simplification helps in better understanding and solving mathematical problems by making equations easier to handle.

In this specific exercise, the expression \((\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5})\) is simplified by first applying the difference of squares formula. This step simplifies the multiplication of conjugates down to a simple subtraction: \(\sqrt{3}^2 - \sqrt{5}^2\). As shown:

• This becomes: \(3 - 5 = -2\)

The beauty of this process is that it removes the complexity of having square roots in the equation and provides a clear, integer result. Mastering these simplification techniques is vital, as it lays the groundwork for tackling more complex algebraic expressions.