Problem 3 - Algebra for College Students

Question

Multiply and simplify where possible. \((3 \sqrt{3})(2 \sqrt{6})\)

Step-by-Step Solution

Answer

The simplified form is \(18 \sqrt{2}\).

Step 1: Multiply the Square Roots

The given expression is \(3 \sqrt{3} imes 2 \sqrt{6}\). First, multiply the coefficients outside the square roots: \(3 \times 2 = 6\). Then, multiply the square roots: \(\sqrt{3} \times \sqrt{6} = \sqrt{18}\).

Step 2: Simplify the Square Root

The result of the multiplication gives us \(6 \sqrt{18}\). Now, simplify \(\sqrt{18}\). Since \(18 = 9 \times 2\) and \(9\) is a perfect square, \(\sqrt{18} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\).

Step 3: Final Simplification

Substitute the simplified square root back into the expression: \(6 \sqrt{18} = 6 \times 3\sqrt{2} = 18 \sqrt{2}\). Therefore, the product \(3 \sqrt{3} \times 2 \sqrt{6}\) simplifies to \(18 \sqrt{2}\).

Key Concepts

Simplifying Square Roots

When you simplify square roots, you are reorganizing the expression to make it simpler and easier to understand. Square roots that involve perfect squares can be broken down further.
For example, if you have the square root of 18, you can simplify it by finding the factors of 18 that include a perfect square.

• 18 can be factored into 9 and 2.
• Here, 9 is a perfect square, because its square root is a whole number: \( \sqrt{9} = 3 \).
• Thus, \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \).

Breaking numbers into components with perfect squares allows you to simplify the expression effectively. This approach makes it easier to deal with calculations involving square roots.

Radical Expressions

A radical expression includes values that are under a radical sign, typically a square root. These expressions can combine numbers and variables.
When simplifying radical expressions, the goal is to remove any factors that are perfect squares from underneath the square root.

• When you have a term like \( \sqrt{x^2 \cdot y} \), it can be simplified to \( x\sqrt{y} \), assuming \( x \) is non-negative.
• This approach cuts down on complexity and makes calculations with these expressions more manageable.
• You can also combine like terms if radical expressions share the same radical part. For example, \( 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \).

Understanding how to manage radical expressions is crucial for handling more complex algebraic expressions.

Multiplication of Radicals

When multiplying radicals, the process is straightforward: multiply the coefficients and the radicands separately before simplifying.
Consider the multiplication of two radical terms: \((3\sqrt{3})(2\sqrt{6})\).

• First, multiply the coefficients: 3 and 2, which gives 6.
• Then, multiply the radicands: \(\sqrt{3} \times \sqrt{6} = \sqrt{18} \).
• Simplify the result if possible: \(6\sqrt{18} \).

As shown in the simplification step, \(\sqrt{18} \) can be reduced to \(3\sqrt{2} \). Finally, multiply back: \(6 \times 3\sqrt{2} = 18\sqrt{2}\).
This method of breaking down and simplifying each part as you go helps make complex radical multiplications easier to solve.