Problem 35 - Algebra 2

Question

Simplify. \((-3 \sqrt{24})(5 \sqrt{20})\)

Step-by-Step Solution

Answer

The expression simplifies to \( -60 \sqrt{30} \).

Step 1: Break Down the Radicals

The first step is to simplify each square root by factoring under the radical sign. We start with \( \sqrt{24} \) and \( \sqrt{20} \).For \( \sqrt{24} \):\[ 24 = 4 \cdot 6 = 2^2 \cdot 6 \]So, \( \sqrt{24} = \sqrt{2^2 \cdot 6} = 2\sqrt{6} \).For \( \sqrt{20} \):\[ 20 = 4 \cdot 5 = 2^2 \cdot 5 \]So, \( \sqrt{20} = \sqrt{2^2 \cdot 5} = 2\sqrt{5} \).

Step 2: Substitute the Simplified Radicals

Replace the original radicals with their simplified forms in the expression:\( (-3 \times 2 \sqrt{6})(5 \times 2 \sqrt{5}) \).

Step 3: Multiply the Constants

Now multiply the constants outside the square roots:\( (-3 \times 2) \) and \( (5 \times 2) \) gives:\((-6)(10) = -60\).

Step 4: Multiply the Radicals

Multiply the radicals together:\( \sqrt{6} \times \sqrt{5} = \sqrt{6 \times 5} = \sqrt{30} \).

Step 5: Combine the Results

Combine the products of the constants and radicals to get the final simplified expression:\( -60 \sqrt{30} \).

Key Concepts

Square Roots

When diving into square roots, you're essentially looking for a number which, when multiplied by itself, results in the original number under the root. For instance, the square root of 9, denoted as \( \sqrt{9} \), results in 3, because 3 multiplied by itself is 9. Simplifying square roots requires breaking down the number under the root into factors, particularly focusing on perfect squares.

• A perfect square is a number like 4, 9, or 16 because it is the product of an integer with itself.
• For example, the number 24 under a square root can be factored to 4 and 6, where 4 is a perfect square (since \( 2 \times 2 = 4 \)). Thus, \( \sqrt{24} \) simplifies to \( 2\sqrt{6} \).

Breaking down numbers into their prime factors can also help identify perfect squares. This simplifies the process, ensuring any perfect square factors are outside the radical.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. They can range from simple expressions like \( 2x + 5 \) to more complex combinations such as \( (-3 \sqrt{24})(5 \sqrt{20}) \). The beauty of algebra is its structured approach to solving problems through systematic manipulation of these expressions.

• In the expression \( (-3 \sqrt{24})(5 \sqrt{20}) \), note how the numeric and radical components are handled separately.
• This separation allows simplification through multiplication of constants and radicals individually, which is typical in algebraic manipulations.

By multiplying constants first and then dealing with the radicals, we maintain order and achieve simplified results accurately.

Radical Expressions

Radical expressions involve roots, symbols that indicate an operation opposite of exponentiation. The square root is a common radical, depicted as \( \sqrt{} \). When simplifying radical expressions, focus on simplifying each component, maintaining a clear distinction between constants and roots.

• Start by rewriting radicals in their simplest form by identifying and extracting perfect squares.
• For example, simplify \( \sqrt{24} \) to \( 2\sqrt{6} \), and \( \sqrt{20} \) to \( 2\sqrt{5} \).

Once simplified, it’s crucial to tackle both constants and radicals systematically. In the context of \( (-3 \sqrt{24})(5 \sqrt{20}) \), multiply constants separately and radicals separately, combining in the end to simplify to \( -60 \sqrt{30} \). This methodical approach aids in controlling the complexity and ensuring clarity.