Problem 26 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt{5}(1-\sqrt{10}) $$

Step-by-Step Solution

Answer

\( \sqrt{5} - 5\sqrt{2} \)

Step 1: Distribute the Square Root

Use the distributive property to distribute the square root of 5 across the expression in the parenthesis. This means multiplying \( \sqrt{5} \) with both 1 and \( -\sqrt{10} \).

Step 2: Multiply Square Roots

Calculate \( \sqrt{5} \times 1 = \sqrt{5} \). Next, calculate \( \sqrt{5} \times -\sqrt{10} = -\sqrt{50} \).

Step 3: Simplify the Radicals

Simplify \( \sqrt{50} \). Note that \( 50 = 25 \times 2 \), so \( \sqrt{50} = \sqrt{25 \times 2} \). This simplifies to \( \sqrt{25} \times \sqrt{2} \), which is \( 5\sqrt{2} \).

Step 4: Write the Simplified Expression

Combine the simplified expressions from the previous steps: \( \sqrt{5} - 5\sqrt{2} \).

Key Concepts

Understanding the Distributive Property

The distributive property is a crucial concept in algebra that allows us to handle expressions by spreading or distributing components across parenthetical terms. This technique is particularly useful when coping with radicals or terms inside parentheses. In our given exercise, we need to distribute the square root of 5, denoted as \( \sqrt{5} \), over the expression \( 1 - \sqrt{10} \).

Utilizing the distributive property, you multiply \( \sqrt{5} \) first with the number 1 and then with \( -\sqrt{10} \). Here's how it breaks down:

• First, distribute \( \sqrt{5} \times 1 \), which simply gives \( \sqrt{5} \).
• Next, distribute \( \sqrt{5} \times -\sqrt{10} \), processing both the numerical values and the radicals involved.

This strategic distribution lays the groundwork for further mathematical operations such as multiplication and simplification.

Square Roots Multiplication Explained

Multiplying square roots might seem daunting at first, but it's quite straightforward once you understand the basic rules. The square root multiplication rules allow us to deal directly with numbers under the radical sign. These rules state that the product of two square roots is the square root of their product.

In our example, multiplying \( \sqrt{5} \) and \( \sqrt{10} \) follows these steps:

• Multiply the numbers under the square roots: \( 5 \times 10 = 50 \).
• The result is \( \sqrt{50} \).
• Always remember to carry forward any minus signs, resulting in \( -\sqrt{50} \).

This operation simply leverages the rule \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \), making it a reliable method for simplifying complex expressions.

Radical Simplification Made Easy

Once we've multiplied the square roots, as seen in the previous sections, the next step involves simplifying these radicals. Simplifying radicals is essential for expressing the result in its simplest form. A radical expression can often be streamlined by factoring out perfect squares.

To simplify \( \sqrt{50} \), we recognize that 50 can be further reduced:

• Break down 50 into its factors: \( 25 \times 2 \).
• Since 25 is a perfect square, we take \( \sqrt{25} = 5 \).
• This leaves us with \( 5\sqrt{2} \).

Using this method, any complex radical can be simplified by identifying and extracting perfect squares, allowing you to express the overall product in an elegantly simplified form, such as in this expression: \( \sqrt{5} - 5\sqrt{2} \).