Problem 45 - Elementary Algebra

Question

Simplify \((3+\sqrt{5})(4-\sqrt{5})\)

Step-by-Step Solution

Answer

Question: Simplify the expression \((3+\sqrt{5})(4-\sqrt{5})\). Answer: The simplified expression is \(7+\sqrt{5}\).

Step 1: Apply FOIL method for multiplying binomials

To multiply the two binomials, we'll apply the FOIL method. Multiply the first term of the first binomial with the first term of the second binomial, the outer terms, the inner terms, and the last terms. \((3+\sqrt{5})(4-\sqrt{5}) = (3 \times 4) + (3 \times -\sqrt{5}) + (\sqrt{5} \times 4) + (\sqrt{5} \times -\sqrt{5})\)

Step 2: Perform the multiplication

Now, multiply the terms and simplify: \((3 \times 4) + (3 \times -\sqrt{5}) + (\sqrt{5} \times 4) + (\sqrt{5} \times -\sqrt{5}) = 12 - 3\sqrt{5} + 4\sqrt{5} - 5\)

Step 3: Combine like terms

We can now combine the like terms, -3√5 and 4√5, and the constants 12 and -5: \(12 - 3\sqrt{5} + 4\sqrt{5} - 5 = (12 - 5) + (-3\sqrt{5} + 4\sqrt{5})\)

Step 4: Simplify the expression

Finally, simplify the expression to get the final answer: \((12 - 5) + (-3\sqrt{5} + 4\sqrt{5}) = 7+\sqrt{5}\) So, \((3+\sqrt{5})(4-\sqrt{5})=7+\sqrt{5}\).

Key Concepts

FOIL Method

When multiplying two binomials, the FOIL method is a commonly used technique that stands for First, Outside, Inside, Last. This acronym represents the order in which you multiply the terms of each binomial.

Take for example the expression \( (3+\sqrt{5})(4-\sqrt{5}) \). First, you multiply the First terms from each binomial: \( 3 \times 4 \). Next, you multiply the Outside terms: \( 3 \times -\sqrt{5} \). Then, the Inside terms are multiplied together: \( \sqrt{5} \times 4 \). Finally, we multiply the Last terms: \( \sqrt{5} \times -\sqrt{5} \). By following this order, you can systematically approach the multiplication without missing any terms.

Multiplying Binomials

When two binomials are multiplied together, every term in the first binomial is multiplied by every term in the second binomial. This can also be seen as an application of the distributive property twice.

Using the example \( (3+\sqrt{5})(4-\sqrt{5}) \), we distribute each term of the first binomial across the second binomial. This would look like:\[ (3 \times 4) + (3 \times -\sqrt{5}) + (\sqrt{5} \times 4) + (\sqrt{5} \times -\sqrt{5}) \]. It is essential to multiply each term correctly and make sure that the signs of the terms are taken into account to avoid any mistakes.

Combining Like Terms

In algebra, combining like terms is a fundamental process that simplifies expressions. Like terms are terms that have the exact same variables raised to the same power, even though they may have different coefficients. After using the FOIL method, we often get like terms that we can combine.

For instance, after performing the FOIL method on \( (3+\sqrt{5})(4-\sqrt{5}) \), we get terms like \( -3\sqrt{5} \) and \( 4\sqrt{5} \). These are like terms because they have the same radical part, \(\sqrt{5}\), and can therefore be combined. Similarly, any constant terms (without variables) can also be combined to simplify the expression further.

Radicals in Algebra

Radicals, often known as roots, are expressions that involve the \(\sqrt[\text{n}]{...}\) symbol, where 'n' represents the degree of the root. In the case of our problem, we are dealing with square roots, which are radicals where \(n=2\). When combining terms with radicals, it's important to remember that only like radicals (those with the same radicand and degree) can be combined.

In the simplified example \( (3+\sqrt{5})(4-\sqrt{5}) \), the terms \( -3\sqrt{5} \) and \( 4\sqrt{5} \) combine as they both involve the square root of 5. Radicals can also be manipulated algebraically, such as when \( \sqrt{5} \times -\sqrt{5} \) simplifies to \(-5\), because the square root of a number, times itself, is just the number.