Problem 301 - Elementary Algebra

Question

In the following exercises, simplify. $$ (4 \sqrt{6})(-\sqrt{18}) $$

Step-by-Step Solution

Answer

-24 \sqrt{3}

Step 1 - Simplify the radicals

Simplify the square roots within the expression. Note that \( \sqrt{18} = \sqrt{9 \times 2} = 3 \sqrt{2} \), so the expression becomes: \( (4 \sqrt{6})(-3 \sqrt{2}) \)

Step 2 - Multiply the constants

Multiply the constant terms together: \( 4 \times -3 = -12 \). So the expression now is: \( -12 (\sqrt{6} \sqrt{2}) \)

Step 3 - Multiply the radicals

Multiply the square roots. Recall that \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). Thus, \( \sqrt{6} \times \sqrt{2} = \sqrt{12} \). The expression now is: \( -12 \sqrt{12} \)

Step 4 - Simplify the final radical

Simplify \( \sqrt{12} \). Note that \( \sqrt{12} = \sqrt{4 \times 3} = 2 \sqrt{3} \). Therefore, the expression becomes: \( -12 \times 2 \sqrt{3} = -24 \sqrt{3} \)

Key Concepts

Radical Simplification

Radical simplification is the process of reducing a radical expression to its simplest form. This involves breaking down the radicand (the number under the radical symbol) into its prime factors and then simplifying. For instance, let's break down the radicand in the given exercise: \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\). This step helps us to understand and simplify more complex expressions by dealing with smaller and simpler parts. Remember: prime factorization of the radicand will always lead you to a simplified radical.

Multiplying Radicals

Multiplying radicals can be straightforward if you follow a few key rules. When you multiply two square roots, you multiply the numbers inside the radicals together. For example, in our exercise: \(\sqrt{6} \times \sqrt{2} = \sqrt{6 \times 2} = \sqrt{12}\). It's crucial to simplify each radical first, if possible, before multiplying to make the process easier and more accurate. If you remember that \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\), you will handle these types of problems with more ease.

Algebraic Expressions

An algebraic expression can include numbers, variables, and operators (like addition and multiplication). When dealing with radicals within algebraic expressions, it's important to handle each part step-by-step. In the given exercise: \( (4\sqrt{6})(-3\sqrt{2}) \), we first simplify and multiply the constants: \(4 \times -3 = -12\). Then, we multiply and simplify the radicals: \(\sqrt{6} \times \sqrt{2} = \sqrt{12} = 2\sqrt{3}\). Finally, our algebraic expression becomes: \(-12 \times 2\sqrt{3} = -24\sqrt{3}\). Following order of operations and proper handling of the constants and radicals is essential.

Elementary Algebra

Elementary algebra forms the basis of all algebraic calculations, including working with radicals. It includes operations like addition, subtraction, multiplication, and division of algebraic expressions. In our exercise, working with constants alongside radicals involves basic multiplication: \( 4 \times -3 = -12 \). Then, we extend this to more complex operations involving radicals. Getting comfortable with these basics ensures a strong foundation for more advanced algebraic concepts. Breaking down problems into smaller steps helps manage more complicated expressions.