Problem 72 - Algebra 2

Question

Multiply. \((4+2 \sqrt{3})(6-3 \sqrt{3})\)

Step-by-Step Solution

Answer

The product of \((4+2 \sqrt{3})(6-3 \sqrt{3})\) is 6.

Step 1: Apply the FOIL method

Start by multiplying the two first terms together, followed by the outer terms, then the inner terms, and finally, the last terms. The FOIL method can be written as follows: \((4+2 \sqrt{3})(6-3 \sqrt{3})\) = \(4*6 + 4*(-3\sqrt{3}) + 2\sqrt{3}*6 + 2\sqrt{3}*(-3\sqrt{3})\)

Step 2: Simplify the terms

Now, simplify each of the terms from the FOIL process: = \(24 - 12\sqrt{3} + 12\sqrt{3} - 18\)

Step 3: Combine like terms

Combine the like terms: = \(24 - 18 = 6\)

Key Concepts

Multiplying Binomials

When multiplying binomials, the FOIL method comes in handy. FOIL stands for First, Outer, Inner, Last. This is a systematic way of multiplying each term in one binomial by each term in the other binomial. Let's illustrate this with the binomials

• First: Multiply the first terms in each binomial: \(4 \times 6 = 24\).
• Outer: Multiply the outer terms: \(4 \times (-3\sqrt{3}) = -12\sqrt{3}\).
• Inner: Multiply the inner terms: \(2\sqrt{3} \times 6 = 12\sqrt{3}\).
• Last: Multiply the last terms: \(2\sqrt{3} \times (-3\sqrt{3}) = -18\).

After applying the FOIL method, you combine all these products to get: \[24 - 12\sqrt{3} + 12\sqrt{3} - 18\]Make sure to handle both the numerical values and the radicals properly when performing each step of multiplication.

Simplifying Radical Expressions

Simplifying radical expressions is an essential skill in many algebraic problems. In our example, after applying FOIL, you end up with terms like \(-12\sqrt{3}\) and \(12\sqrt{3}\). Simplifying these requires recognizing and combining like terms.
When two radicals are identical, you can treat them like variables or numbers in terms of addition and subtraction:

• For example, \(-12\sqrt{3} + 12\sqrt{3}\) results in \(0\sqrt{3}\), which simplifies to 0.

Always ensure that you simplify terms without altering their original values. In cases where radicals don’t match, they cannot be combined, much like how you can’t add "apples" to "oranges." Additionally, simplifying radical terms by squaring them, as done with \(2\sqrt{3} \times (-3\sqrt{3})\), helps eliminate the radical, seen here as: \((-3\sqrt{3})^2 = -18\). This results because \((\sqrt{3})^2\) equals 3.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operational symbols. In our example, the expression \((4 + 2\sqrt{3})(6 - 3\sqrt{3})\) is a product of two binomials. We use the FOIL method to expand it and simplify the expression further.
Terms inside an algebraic expression can include integers, fractions, and radicals. When expanding, you must be mindful of handling these various elements correctly.
After multiplying and simplifying using the FOIL method, the result becomes: \(24 - 18\)

• The radical expressions have already been simplified to 0, as they are canceled out.
• Finally, simplifying the entire expression, we find that it equals 6.

Understanding how to thoroughly simplify and combine like terms will aid in solving more complicated algebraic problems.