Problem 8 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{3}(5 \sqrt{3}-2 \sqrt{6}) $$

Step-by-Step Solution

Answer

The simplified expression is \( 15 - 6\sqrt{2} \)

Step 1: Simplify Inside Parenthesis

First, take the square root of 3 and multiply it by each term inside the parenthesis. Hence, \( \sqrt{3} \times 5 \sqrt{3} \) yields \( 5*3 \) and \( \sqrt{3} \times 2\sqrt{6} \) becomes \( 2\sqrt{18} \).

Step 2: Simplify the Resulting Terms

\(\sqrt{18} \) can be simplified to \( 3\sqrt{2} \) because \( 18 = 9 \times 2 = (3^2) \times 2 \) and square root of \( 3^2 \) is 3. So, \( 2\sqrt{18} \) becomes \( 2 \times 3\sqrt{2} = 6 \sqrt{2} \).

Step 3: Prune the Expression

After simplifying all terms, the expression becomes \( 15 - 6\sqrt{2} \)

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves reducing expressions to their simplest form, making them easier to work with or solve.
In the given problem, the goal is to simplify the expression \( \sqrt{3}(5 \sqrt{3}-2 \sqrt{6}) \). This process consists of various steps:

• Distributing a common factor across terms within parentheses, as in the problem where \( \sqrt{3} \) is distributed to both \( 5 \sqrt{3} \) and \( -2 \sqrt{6} \).
• Combining like terms and simplifying any resulting radicals or expressions.

Carefully following these procedures ensures the expression becomes clearer and simpler, making further mathematical operations more manageable.

Square Roots

Square roots are a fundamental concept in mathematics. They represent a number which, when multiplied by itself, yields the original number. The symbol for a square root is \( \sqrt{} \).
For example, \( \sqrt{9} \) results in 3 since \( 3 \times 3 = 9 \).
This concept is used in the given exercise where \( \sqrt{18} \) is simplified to \( 3\sqrt{2} \).

• Here, understanding that 18 can be factored into \( 9 \times 2 \) aids in recognizing \( 9 \) as a perfect square, simplifying \( \sqrt{18} \) to \( \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \).

This process helps break down more complex numbers into manageable factors, aiding in the simplification of radical expressions.

Multiplication of Radicals

Multiplying radicals involves combining them through multiplication, keeping under the radical sign as long as possible.
In the exercise, \( \sqrt{3} \times 5\sqrt{3} \) simplifies by utilizing the property \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \).

• This results in \( \sqrt{3 \times 3} = \sqrt{9} = 3 \), thus when multiplied by 5, the expression becomes 15.
• Similarly, \( \sqrt{3} \times \sqrt{6} = \sqrt{18} \) is simplified to \( 3\sqrt{2} \).

The important rules for these operations include keeping similar radicands together and simplifying further by factoring out perfect squares. Recognizing and applying these principles makes multiplying radicals a straightforward task.