Problem 20 - Elementary Algebra

Question

Multiply. (Assume all variables are non negative.) $$ 18--\sqrt{3} \cdot 6 \sqrt{3} $$

Step-by-Step Solution

Answer

The solution is 36.

Step 1: Understanding the Expression

The expression given is \(18 - (-\sqrt{3}) \cdot 6 \sqrt{3}\). Here, we need to multiply \(-\sqrt{3}\) with \(6\sqrt{3}\) and then subtract the result from 18.

Step 2: Simplify the Multiplication of Radicals

We calculate \((-\sqrt{3}) \cdot 6 \sqrt{3}\). This can be broken down as \(-1 \cdot \sqrt{3} \cdot 6 \cdot \sqrt{3}\) or equivalently \(-1 \cdot 6 \cdot \sqrt{3} \cdot \sqrt{3}\).

Step 3: Apply the Rule of Multiplying Radicals

To multiply \(\sqrt{3}\) by \(\sqrt{3}\), use the rule \(\sqrt{a} \cdot \sqrt{a} = a\). Hence, \(\sqrt{3} \cdot \sqrt{3} = 3\).

Step 4: Complete the Multiplication

Now, substitute back \(\sqrt{3} \cdot \sqrt{3} = 3\) into the expression: \(-1 \cdot 6 \cdot 3 = -18\).

Step 5: Subtract from 18

The original expression is \(18 - (-18)\). Subtracting \(-18\) is the same as adding 18, hence it becomes \(18 + 18\).

Step 6: Calculating the Final Result

Evaluate the expression \(18 + 18 = 36\). The solution to the original problem is thus 36.

Key Concepts

Understanding Basic Mathematical Operations

When dealing with the multiplication and subtraction of numbers, it's important to remember the rules governing basic mathematical operations. In our given problem, we must carry out multiplication and then subtraction.

Key points to consider include:

• Multiplication: This is where you combine numbers to get a larger product. It can involve numbers, as well as variables and expressions like radicals.
• Negative Sign: Multiplying a negative value changes the sign of your result. So, \(-1 \cdot a\) is \(-a\).
• Subtraction: In expressions like \(a - (-b)\), subtracting a negative is the same as adding, or \(a + b\).

Understanding these operations ensures we master more complex expressions, setting the foundation for algebraic simplification.

Delving into Radical Expressions

Radical expressions involve roots, such as square roots. They appear frequently in algebra and need careful handling.

Some key features include:

• Radicals represent roots, such as \(\sqrt{3}\), which is a square root.
• To multiply radicals, you can use the properties of exponents. If same bases are involved, like \(\sqrt{a} \cdot \sqrt{a}\), it simplifies to \(a\).
• When multiplying a radical by a non-radical, treat the radical as a separate entity unless simplified further.

For example, multiplying \(-\sqrt{3} \cdot 6\sqrt{3}\) allows us to use the property \(\sqrt{3} \cdot \sqrt{3} = 3\) to simplify the expression neatly.

Steps of Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. It involves combining like terms and applying known mathematical rules.

Here's how we simplify within our exercise:

• Start by simplifying within groupings such as parentheses or radicals.
• Multiply terms together, ensuring that rules like multiplying radicals are applied correctly.
• Combine any like terms; in this exercise, turn \(-18\) from the multiplication into a positive addition with \(18\), due to double negation.

The goal is to reach the simplest form, which in this case results in \(36\), demonstrating clear and streamlined algebraic work.