Problem 15 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt{3} \sqrt{15} $$

Step-by-Step Solution

Answer

\(\sqrt{3} \sqrt{15} = 3\sqrt{5}\).

Step 1: Review the Multiplication Rule for Square Roots

Start by understanding that when you multiply two square roots together, you can combine them under one square root. If you have \(\sqrt{a} \times \sqrt{b}\), it simplifies to \(\sqrt{a \times b}\).

Step 2: Apply the Multiplication Rule

Apply the rule to \(\sqrt{3}\) and \(\sqrt{15}\): \[\sqrt{3} \times \sqrt{15} = \sqrt{3 \times 15}.\]

Step 3: Simplify the Expression

Calculate the product under the square root: \[3 \times 15 = 45.\]So, it becomes \(\sqrt{45}\).

Step 4: Simplify the Square Root Further

The next step is to simplify \(\sqrt{45}\). Recognize 45 as \(9 \times 5\), where 9 is a perfect square. Rewrite it as: \[\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}.\]This simplifies to \(3\sqrt{5}\) because \(\sqrt{9} = 3\).

Key Concepts

Multiplication of Square Roots

Understanding the multiplication of square roots is a foundational algebraic operation that helps simplify complex expressions. When we multiply two square roots, such as \(\sqrt{a}\) and \(\sqrt{b}\), they can be combined into one square root:

• \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)

This property is extremely useful when dealing with different square roots since it allows us to collapse them into a single expression, potentially simplifying calculations. For example, multiplying \(\sqrt{3}\) by \(\sqrt{15}\) combines to \(\sqrt{3 \times 15}\), which simplifies into \(\sqrt{45}\). It makes it easier to proceed further with simplifications and calculations. This rule holds true as long as all values involved are positive, ensuring the expression remains within the realm of real numbers.

Simplifying Square Roots

Simplifying square roots involves reducing them to their most basic form where possible. This process often requires identifying factors that are perfect squares. Here's a straightforward approach to simplifying a square root, such as \(\sqrt{45}\):

• Find factors of the number under the square root that include a perfect square.
• In the case of 45, it can be factored into 9 and 5, where 9 is a perfect square.

Rewrite the square root expression as a product of two square roots:

• \(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}\)

Since \(\sqrt{9} = 3\), the simplification results in \(3\sqrt{5}\). This final form is more straightforward and elegant, especially in algebraic manipulations or solving equations.

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. Recognizing perfect squares is crucial when working with square roots, as they simplify square root calculations significantly by reducing complex square root expressions. Some common perfect squares include:

• 1 (\(1^2\))
• 4 (\(2^2\))
• 9 (\(3^2\))
• 16 (\(4^2\))
• 25 (\(5^2\))

When simplifying square roots, look for these numbers within the factors of any expression under the square root. For instance, in \(\sqrt{45}\), the number 9 is identified as a perfect square (since \(3^2 = 9\)), thereby simplifying the expression dramatically. The process becomes more efficient when familiar with perfect squares, making them an essential tool for simplifying square roots easily.