Question

Find a conjugate of each expression and the product of the expression with the conjugate. $$ -\sqrt{5}-\sqrt{6} $$

Short answer

Answer: The complex conjugate of \(-\sqrt{5}-\sqrt{6}\) is \(-\sqrt{5}+\sqrt{6}\), and the product of the expression and its conjugate is \(11\).

Step-by-step solution

Identify the Expression

The given expression is: $$ -\sqrt{5}-\sqrt{6} $$

Find the Conjugate

To find the conjugate, change the sign of the \(\sqrt{6}\) term. The conjugate of the given expression is: $$ -\sqrt{5}+\sqrt{6} $$

Calculate the Product of the Expression and Its Conjugate

Now, we will calculate the product of the original expression and its conjugate. $$ (-\sqrt{5}-\sqrt{6})(-\sqrt{5}+\sqrt{6}) \\ $$ We can use the FOIL (first, outer, inner, last) method to calculate the product. $$ = (-\sqrt{5} \times -\sqrt{5}) + (-\sqrt{5} \times \sqrt{6})+ (-\sqrt{6} \times -\sqrt{5})+ (-\sqrt{6} \times \sqrt{6}) $$ Simplify: $$ = 5 -(-\sqrt{30}+\sqrt{30})+ 6 $$ The middle terms cancel out, so our final answer is: $$ = 5 + 6 = 11 $$ In conclusion, the conjugate of the given expression is \(-\sqrt{5}+\sqrt{6}\), and the product of the expression and its conjugate is \(11\).

Key concepts

Square Roots

Square roots are mathematical figures that represent one of two equal factors of a number. For instance, when you have the square root of 25, it means you're looking for a number that, when multiplied by itself, gives 25. That number is 5. Square roots are represented by the radical symbol \( \sqrt{\cdot} \).Understanding how to work with square roots becomes crucial when dealing with expressions. In the given exercise, we have expressions involving square roots like \( \sqrt{5} \) and \( \sqrt{6} \). Each of these represents a number that multiplied by itself gives 5 and 6, respectively.
Manipulating square roots involves several properties:

• \( \sqrt{a^2} = a \): The square root of a squared number gives the original number.
• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \): Multiplying square roots is the same as taking the product of the numbers inside the root.
• \( \sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}} \): Division inside a square root can be written as a division of square roots.

These properties are essential in simplifying expressions, especially when they involve operations like addition, subtraction, and multiplication as seen in the exercise here.

FOIL Method

The FOIL method is a technique used to multiply two binomials – expressions that consist of two terms each. Understanding this concept is crucial when finding the product of the given expression and its conjugate. In our exercise, this would be multiplying \((-\sqrt{5} - \sqrt{6})\) by its conjugate, \((-\sqrt{5} + \sqrt{6})\).
The term 'FOIL' stands for:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

In our example, using FOIL:- **First**: \((-\sqrt{5})\times(-\sqrt{5}) = 5\)- **Outer**: \((-\sqrt{5})\times(\sqrt{6}) = -\sqrt{30}\)- **Inner**: \((-\sqrt{6})\times(-\sqrt{5}) = \sqrt{30}\)- **Last**: \((-\sqrt{6})\times(\sqrt{6}) = 6\)The key here is that the outer and inner products cancel each other out. This is a common occurrence when multiplying an expression by its conjugate, leading to simpler results.

Product of Expressions

The product of expressions involves multiplying two algebraic expressions together. The process not only combines the terms but also simplifies them where possible. In the exercise given, we multiply \((-\sqrt{5} - \sqrt{6})\) by its conjugate. The use of the FOIL method shows this step-by-step.
Here's how the product works in more detail:1. **Identify the terms**: Each component of the expression is individually multiplied.2. **Simplify using algebraic rules**: Identify any common terms or factors.3. **Cancel out common terms**: As shown in the solution, the middle terms cancel out.4. **Sum the remaining terms**: This will give the final simplified product.
In this particular problem, multiplying the expression \((-\sqrt{5} - \sqrt{6})\) by its conjugate \((-\sqrt{5} + \sqrt{6})\) simplifies to 11. This is because the expression of the form \((a-b)(a+b)\) equals \(a^2 - b^2\). Therefore, it becomes a straightforward calculation of \(5 + 6 = 11\). Understanding these steps helps simplify complex multiplication and is a fundamental part of algebra.