Problem 62 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (\sqrt{6}-4 \sqrt{2})(3 \sqrt{6}+\sqrt{2}) $$

Step-by-Step Solution

Answer

The simplified expression is \(10 - 22\sqrt{3}\).

Step 1: Identify the expression type

The given expression \[(\sqrt{6} - 4\sqrt{2})(3\sqrt{6} + \sqrt{2})\] is in the form of \((a - b)(c + d)\) which suggests using the distributive property (also known as the FOIL method: First, Outer, Inner, Last) for multiplication.

Step 2: Apply FOIL method

To multiply the two binomials, we use the FOIL method:- **First**: Multiply the first terms: \((\sqrt{6}) \times (3\sqrt{6}) = 3 \times 6 = 18\).- **Outer**: Multiply the outer terms: \((\sqrt{6}) \times (\sqrt{2}) = \sqrt{12} = 2\sqrt{3}\).- **Inner**: Multiply the inner terms: \((-4\sqrt{2}) \times (3\sqrt{6}) = -12\sqrt{12} = -24\sqrt{3}\).- **Last**: Multiply the last terms: \((-4\sqrt{2}) \times (\sqrt{2}) = -8\).

Step 3: Combine like terms

Now, add all the terms together:\[18 + 2\sqrt{3} - 24\sqrt{3} - 8 = 18 - 22\sqrt{3} - 8\].Combine the constant terms:\[18 - 8 = 10\],and simplify the square root terms:\[2\sqrt{3} - 24\sqrt{3} = -22\sqrt{3}\].Thus, the simplified expression is:\[10 - 22\sqrt{3}\].

Key Concepts

Simplifying Radicals

Simplifying radicals is an essential skill when dealing with radical expressions. A radical expression includes a square root, a cube root, or any other root. The primary aim of simplifying radicals is to express them in their simplest form. For example, if you come across a term like \( \sqrt{12} \), you want to break it down into simpler square roots. Here, you would factor 12 into 4 and 3, resulting in \( \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \). The idea is to break down numbers under the radical sign into perfect squares. This involves:

• Identifying perfect square factors
• Taking roots of these perfect squares outside the radical
• Leaving any unfactorable reminders within the radical

This process helps in making the arithmetic straightforward and tidier and also facilitates combining similar terms efficiently when further operations are required.

FOIL Method

The FOIL method is a memorable technique used specifically for multiplying two binomials. The name "FOIL" stands for First, Outer, Inner, and Last, which are the pairs of terms we multiply together in sequence. It's like a step-by-step recipe:

• First: Multiply the first terms in each binomial. For example, if we have \((a + b)(c + d)\), we multiply \(a \times c\).
• Outer: Multiply the outer terms, i.e., \(a \times d\).
• Inner: Multiply the inner terms, i.e., \(b \times c\).
• Last: Multiply the last terms in each binomial, i.e., \(b \times d\).

After calculating these four products, sum them up to get your final result. For instance, in the expression \((\sqrt{6} - 4\sqrt{2})(3\sqrt{6} + \sqrt{2})\), applying FOIL helps manage the multiplication of complex terms with ease and ensures none of the terms are overlooked.

Distributive Property

The distributive property, often encountered in algebra, is a useful rule that illustrates how multiplication distributes over addition or subtraction. It's expressed as \(a(b + c) = ab + ac\). Essentially, it allows you to "distribute" a factor over terms inside a parenthesis. This property is extremely helpful in complicated expressions where direct computation appears challenging.
When applied to binomials like \((\sqrt{6} - 4\sqrt{2})(3\sqrt{6} + \sqrt{2})\), each term within the first binomial is multiplied by every term in the second binomial using distribution. The distributive property is visually recognized as applying the FOIL method, which organizes this process into a systematic multiplication order. This ensures that each component of the binomial is accounted for without missing any possible combination of terms. The resulting expression is simplified by combining like terms.

Binomials Multiplication

Multiplying binomials is a fundamental operation in algebra and it is important to get a clear understanding of how it works. A binomial consists of two distinct terms, for example, \((A + B)\). When multiplying two binomials, such as \((A + B)(C + D)\), you end up with four distinct terms after applying the distributive property or FOIL method.Each pair of terms from the binomials is multiplied as such:

• First terms multiplied: \(A \times C\)
• Outer terms multiplied: \(A \times D\)
• Inner terms multiplied: \(B \times C\)
• Last terms multiplied: \(B \times D\)

After performing these multiplications, you add the results to arrive at a single expression. In the context of the original exercise with radicals, remembering to simplify any radical expressions after multiplying binomials is crucial to maintaining clarity and achieving the simplest form of the expression.