Problem 27 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ 3 \sqrt{5}(4-\sqrt{5}) $$

Step-by-Step Solution

Answer

The simplified expression is \(12\sqrt{5} - 15\).

Step 1: Distribute the Terms

Start by distributing the term outside the parentheses across the terms inside the parentheses. Multiply \(3\sqrt{5}\) by each term inside the parentheses:\[3\sqrt{5} \times 4 - 3\sqrt{5} \times \sqrt{5}\]

Step 2: Multiply the Terms

Calculate the products:- \(3\sqrt{5}\times 4 = 12\sqrt{5}\)- \(3\sqrt{5}\times \sqrt{5} = 3(\sqrt{5} \times \sqrt{5}) = 3 \cdot 5 = 15\)

Step 3: Simplify the Expression

Combine the results from the distribution in step 2: \[12\sqrt{5} - 15\]

Key Concepts

Multiplication of Radicals

When multiplying radicals, it is essential to understand how to properly handle numbers that are under the square root.
Multiplication of the same radical is straightforward: you can simply multiply the numbers inside the square root together. For instance, multiplying \( \sqrt{a} \) with \( \sqrt{b} \) gives \( \sqrt{a \times b} \).

• Example: \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \).

However, when you multiply a radical with a non-radical term, first multiply the constants outside the radical. Then, apply multiplication to any numbers inside the radical signs separately.

• For instance, if you multiply \( 3\sqrt{5} \) by \( 4 \), the constants \(3\) and \(4\) multiply to make \(12\), with the radical remaining as \(\sqrt{5}\).

This concept becomes even more useful when you work with expressions involving multiple terms, as seen when using the distributive property.

Simplifying Expressions

Simplifying expressions is an integral part of working with radicals. It involves reducing the expression to its simplest form, so it is easier to understand and use in further calculations.
In an expression that involves radicals, simplifying often means ensuring that no fraction remains under the radical and the radical itself is in its simplest form.

• For example, \( \sqrt{36} = 6 \) because 36 is a perfect square.

In the case of the exercise, after multiplying \( 3\sqrt{5} \times \sqrt{5} \), you get \( 3 \times 5 = 15 \), which results from the fact that \( \sqrt{5} \times \sqrt{5} = 5 \).
You also need to combine like terms to simplify expressions fully. In our initial problem, after performing multiplication, you arrive at \( 12\sqrt{5} - 15 \). This is already simplified as there are no like terms to combine.

Distributive Property

The distributive property is a helpful mathematical rule that allows you to multiply a single term by two or more terms within parentheses.
This property is especially useful when dealing with algebraic expressions. It states that \( a(b + c) = ab + ac \).
In the given exercise, this was used to distribute \( 3\sqrt{5} \) across the terms in the parenthesis \((4 - \sqrt{5})\).

• First distribute to get \( 3\sqrt{5} \times 4 \) and \( 3\sqrt{5} \times (-\sqrt{5}) \).

By following this step, it becomes easy to effectively multiply and simplify each term separately. Later, each term is calculated to give \(12\sqrt{5} - 15\). Using the distributive property breaks down the expression into manageable pieces, making multiplication by radicals easier to handle.