Problem 78 - A Graphical Approach to College Algebra

Question

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$(4 \sqrt{5}-1)(3 \sqrt{5}+2)$$

Step-by-Step Solution

Answer

The simplified expression is \(58 + 5\sqrt{5}.\)

Step 1: Apply Distributive Property

First, apply the distributive property (also known as FOIL method for binomials) to multiply each term in the first binomial by each term in the second binomial. This means you will perform these multiplications: \((4\sqrt{5}) \times (3\sqrt{5}), (4\sqrt{5}) \times 2, -1 \times (3\sqrt{5}),\) and \(-1 \times 2.\)

Step 2: Calculate Each Product

Perform the multiplication for each pair:1. \((4\sqrt{5}) \times (3\sqrt{5}) = 4 \times 3 \times \sqrt{5} \times \sqrt{5} = 12 \times 5 = 60,\) since \(\sqrt{5} \times \sqrt{5} = 5.\)2. \((4\sqrt{5}) \times 2 = 8\sqrt{5}.\)3. \((-1) \times (3\sqrt{5}) = -3\sqrt{5}.\)4. \((-1) \times 2 = -2.\)

Step 3: Combine Like Terms

Sum up all the resulting terms from Step 2: \(60 + 8\sqrt{5} - 3\sqrt{5} - 2.\) Combine the like terms, which are the radical terms: \(8\sqrt{5} - 3\sqrt{5} = 5\sqrt{5}.\)

Step 4: Write the Simplified Expression

Combine all terms to write the final simplified expression: \(60 - 2 + 5\sqrt{5}.\) This simplifies to: \(58 + 5\sqrt{5}.\)

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to multiply a single term by a sum or difference inside parentheses. This becomes particularly useful in the context of multiplying expressions, especially when dealing with binomials. To apply the distributive property, you multiply each term in one parenthesis by each term in the other. Here's how it works in practice:

Consider the expression

• \((4 \sqrt{5}-1)(3 \sqrt{5}+2)\).

This means you take each part of the first binomial and multiply it by each part of the second binomial. This can be remembered through the FOIL method:

• First: Multiply the first terms of each binomial: \((4\sqrt{5}) \times (3\sqrt{5})\)
• Outer: Multiply the outer terms: \((4\sqrt{5}) \times 2\)
• Inner: Multiply the inner terms: \((-1) \times (3\sqrt{5})\)
• Last: Multiply the last terms: \((-1) \times 2\)

Using the distributive property systematically like this guarantees that every part of the expression gets multiplied correctly and fully.

Simplifying Expressions

Simplifying expressions involves combining like terms and performing operations to reduce the expression to its simplest form. Once you apply the distributive property and make your multiplications, your next goal is to combine all similar or like terms. Like terms are terms that have the same variable raised to the same power. Here, we focused on greatly simplifying the expression:

After applying the distributive property to

• \((4 \sqrt{5}-1)(3 \sqrt{5}+2)\)

and multiplying each pair, we arrive at terms:
- \(60\),
- \(8\sqrt{5}\),
- \(-3\sqrt{5}\),
- and \(-2\).

Identifying like radical terms is essential here:

• \(8\sqrt{5} - 3\sqrt{5} = 5\sqrt{5}\)

This step helps you streamline the expression to

• \(58 + 5\sqrt{5}\)

Simplified expressions are typically more elegant and easier to understand.

Radicals

Radicals involve working with roots, like square roots, which are very common in algebra. A radical expression usually contains a root symbol (√), and simplifying radicals often requires understanding how to manipulate these roots.

In our example,

• \(\sqrt{5}\)

are the square root terms involved. When you multiply a radical by itself, it resolves to just the number under the root symbol, simplifying to a rational number:

• \(\sqrt{5} \times \sqrt{5} = 5\)

Also, it’s crucial to combine radicals only when you have similar terms, much like regular variables such as \(x\).

• For instance, \(8\sqrt{5} - 3\sqrt{5}\) can be combined to give \(5\sqrt{5}\).

Understanding radicals and correctly simplifying them is key in algebra, and with continued practice, manipulating these expressions becomes second nature.