Problem 37 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((2 \sqrt{6}+3 \sqrt{5})(\sqrt{8}-3 \sqrt{12})\)

Step-by-Step Solution

Answer

The simplest radical form is \(8\sqrt{3} - 36\sqrt{2} + 6\sqrt{10} - 18\sqrt{15}\).

Step 1: Use the Distributive Property

To find the product, we distribute each term in the first binomial with each in the second. We calculate: 1. \(2\sqrt{6} \times \sqrt{8} \)2. \(2\sqrt{6} \times (-3\sqrt{12}) \)3. \(3\sqrt{5} \times \sqrt{8} \)4. \(3\sqrt{5} \times (-3\sqrt{12}) \)

Step 2: Calculate Each Term

Calculate each of the terms determined in Step 1:1. \(2\sqrt{6} \cdot \sqrt{8} = 2 \sqrt{48} = 2 \cdot 4\sqrt{3} = 8\sqrt{3}\)2. \(2\sqrt{6} \cdot (-3\sqrt{12}) = -6\sqrt{72} = -6 \cdot 6\sqrt{2} = -36\sqrt{2}\)3. \(3\sqrt{5} \cdot \sqrt{8} = 3\sqrt{40} = 3 \cdot 2\sqrt{10} = 6\sqrt{10}\)4. \(3\sqrt{5} \cdot (-3\sqrt{12}) = -9\sqrt{60} = -9 \cdot 2\sqrt{15} = -18\sqrt{15}\)

Step 3: Combine Like Terms

After simplifying the terms from Step 2:1. \(8\sqrt{3}\)2. \(-36\sqrt{2}\)3. \(6\sqrt{10}\)4. \(-18\sqrt{15}\)These terms are not like terms, so they cannot be combined further. The expression in its simplest radical form is:\(8\sqrt{3} - 36\sqrt{2} + 6\sqrt{10} - 18\sqrt{15}\)

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps us to multiply terms efficiently. When you have two binomials, such as \((a + b)(c + d)\), the distributive property allows you to multiply each term in the first binomial by each term in the second binomial. This is sometimes known as expanding the expression.

To apply the distributive property, simply move through each term systematically:

• First, multiply the first term of the first binomial by each term of the second binomial.
• Second, multiply the second term of the first binomial by each term of the second binomial.

For example, in the expression \((2 \sqrt{6} + 3 \sqrt{5})(\sqrt{8} - 3 \sqrt{12})\), you calculate:

• \(2 \sqrt{6} \times \sqrt{8}\), \(2 \sqrt{6} \times (-3 \sqrt{12})\), \(3 \sqrt{5} \times \sqrt{8}\), \(3 \sqrt{5} \times (-3 \sqrt{12})\).

This way, you ensure that every term from one binomial is multiplied by every term from the other.

Binomial Multiplication

Binomial multiplication goes hand-in-hand with the distributive property. When you're working with binomials, you're essentially dealing with an expression that includes two terms.

When multiplying two binomials, like in the example with radicals, you use the distributive property to expand before simplifying. It's like unfolding a box that has two terms on each side. After expanding, each of the original terms interacts with every other term across the binomials.

For instance, after distributing \((2 \sqrt{6} + 3 \sqrt{5})(\sqrt{8} - 3 \sqrt{12})\), you will have multiple terms: \[2 \sqrt{6} \times \sqrt{8}, \quad 2 \sqrt{6} \times (-3 \sqrt{12}), \quad 3 \sqrt{5} \times \sqrt{8}, \quad 3 \sqrt{5} \times (-3 \sqrt{12}).\] This comprehensive step ensures no term is left out and sets the stage for subsequent simplification.

Radical Simplification

Simplifying radical expressions is essential to make expressions easier to work with. A radical expression typically involves a number within a square root. To simplify it, find the largest perfect square that divides the number under the radical and rewrite it.

For example, consider \(\sqrt{72}\):

• The largest perfect square factor of 72 is 36.
• Rewrite \(\sqrt{72}\) as \(\sqrt{36 \times 2}\).
• This simplifies further to \(6\sqrt{2}\).

In our exercise, each multiplication term from the expansion should be simplified using this method to make it more manageable. Such simplification makes further operations straightforward.

Combining Like Terms

After simplifying radical expressions, the next step is to combine like terms. However, unlike regular addition or subtraction, terms under a square root can only be combined if they contain the same radical part.

In our example, the terms \(8\sqrt{3}, -36\sqrt{2}, 6\sqrt{10},\) and \(-18\sqrt{15}\) each have different radical parts (\(\sqrt{3}\), \(\sqrt{2}\), \(\sqrt{10}\), \(\sqrt{15}\)). This means:

• They are not like terms, and thus, cannot be added or subtracted to yield a simpler expression directly.

When dealing with radicals, always look for like terms—those that have the same number under the root sign. If they match, you can sum or subtract the coefficients. Otherwise, keep them as they are for the simplest form.