Problem 4 - 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. $$ \sqrt{5} \cdot \sqrt{45} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{5} \cdot \sqrt{45} \) is 15.

Step 1: Review the Problem

The problem involves multiplying two square roots: \( \sqrt{5} \) and \( \sqrt{45} \). The goal is to simplify this product to its simplest form.

Step 2: Multiply the Radicands

According to the property of square roots, \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b} \). Therefore, \( \sqrt{5} \cdot \sqrt{45} = \sqrt{5 \times 45} \). Calculate the product inside the square root: \( 5 \times 45 = 225 \). Thus, it becomes \( \sqrt{225} \).

Step 3: Simplify the Square Root

Find the square root of 225. Since 225 is a perfect square, \( \sqrt{225} = 15 \).

Step 4: Write the Final Answer

The product \( \sqrt{5} \cdot \sqrt{45} \) simplifies to 15.

Key Concepts

Square Roots

A square root is a mathematical function represented by the radical symbol \( \sqrt{} \). It finds a number that, when multiplied by itself, results in the original value under the radical, known as the radicand. For instance, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). Square roots are fundamental in simplifying equations and appear frequently in algebra. They can represent both positive and negative roots, though typically we focus on the principal (positive) root in basic math.

• The number inside the square root is called the radicand.
• The square root function essentially "undoes" the squaring of a number.
• Square roots can be simplified by identifying perfect squares.

Understanding square roots also involves knowing the properties of numbers under the radical sign. We often look to "simplify" square roots when possible.

Perfect Square

A perfect square is a number that can be expressed as the product of an integer by itself. Understanding perfect squares is crucial when simplifying square roots because if a number is a perfect square, its square root is an integer. For instance, \( 16 \) is a perfect square since \( 4 \times 4 = 16 \), hence \( \sqrt{16} = 4 \).Perfect squares are not limited to positive integers. They can also be fractions, like \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \). Recognizing these helps in simplifying square roots, especially in problems where we encounter bigger numbers or need to break down radicands:

• Recognize smaller perfect squares: 1, 4, 9, 16, 25, 36, etc.
• Break down larger radicands into products of smaller perfect squares and simplify accordingly.

In the given problem, \( 225 \) is identified as a perfect square, simplifying the solution.

Multiplying Radicals

Multiplying radicals might seem challenging, but it's straightforward once you know the rules. The key property is \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b} \). This means you can multiply the numbers inside the radicals together, then simplify the result if possible. For example, multiplying \( \sqrt{2} \cdot \sqrt{8} \) gives \( \sqrt{16} \), which simplifies to 4.Several important points are key when multiplying radicals:

• Ensure the indices (type of root) are the same; this principle specifically applies to square roots.
• Multiply only the radicands, not the square root symbols themselves.
• After multiplying, check if the new radicand is a perfect square for further simplification.

This concept was applied when combining the radicals \( \sqrt{5} \) and \( \sqrt{45} \) to find \( \sqrt{225} \), simplifying finally to 15.