Problem 9 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression is \(15 - 2\sqrt{18}\)

Step 1: Distributing the terms

Distribute the term \(\sqrt{3}\) with the terms inside the parenthesis. In other words, \(\sqrt{3} \cdot 5\sqrt{3} - \sqrt{3} \cdot 2\sqrt{6}\)

Step 2: Applying the multiplication property of radicals

Apply the multiplication property of radicals \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). So, the expression becomes: \(5\sqrt{3^2} - 2\sqrt{3 \cdot 6}\)

Step 3: Simplifying the expression

Simplify the terms to get the simplest form. \(5\sqrt{9} - 2\sqrt{18} = 5\cdot 3 - 2\sqrt{18} = 15 - 2\sqrt{18}\)

Key Concepts

multiplication property of radicals

When working with radicals, a common operation we perform is multiplication. The multiplication property of radicals helps simplify these tasks. This property states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \). It means that when you multiply two square roots, you can combine them under one radical sign. This property is handy for simplifying expressions that involve the multiplication of radicals.

For example, in our exercise, we had an expression \( \sqrt{3} \cdot 5 \sqrt{3} \). You can use the multiplication property of radicals to combine the two \( \sqrt{3} \) terms by multiplying the numbers under the radicals together: \( \sqrt{3} \cdot \sqrt{3} = \sqrt{3^2} = 3 \). Hence, the expression becomes \( 5 \cdot 3 \).

• Combine the radicals: Multiply the numbers inside the radicals.
• Apply the property: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \).
• Simplify further: Continue reducing until you reach the simplest form.

distributive property

The distributive property is a powerful algebraic tool that allows us to break up and simplify expressions. It lets you multiply a single term across terms inside a parenthesis. This is represented as \( a(b + c) = ab + ac \).

In the context of simplifying expressions with radicals, this property aids in expanding and simplifying the terms. For instance, consider the problem \( \sqrt{3}(5 \sqrt{3} - 2 \sqrt{6}) \). Using the distributive property, \( \sqrt{3} \) multiplies both \( 5 \sqrt{3} \) and \( -2 \sqrt{6} \):

• Multiply: \( \sqrt{3} \cdot 5\sqrt{3} \)
• Then: \( \sqrt{3} \cdot (-2\sqrt{6}) \)

This breaks down the expression so each part becomes manageable by applying subsequent algebraic properties.

The key to mastering this property is recognizing which terms to distribute and practicing with a variety of problems to become comfortable with its applications.

radicals in algebra

Radicals show up in algebra when we deal with roots, typically square roots. They may look complex, but understanding how to work with them is essential. In simplest terms, radical expressions are algebraic expressions that include a square root symbol. Dealing effectively with radicals involves knowing when and how to apply various algebraic rules.

Radicals can often be intimidating due to their appearance. However, they follow systematic rules that make them easier to manage.

• Recognize radicals: Identify presence of square roots.
• Combine using properties: Apply multiplication property of radicals.
• Use underlying algebra rules: Employ distribution or other properties as needed.

In our example from the exercise, radicals like \( \sqrt{3} \) and \( \sqrt{6} \) were involved. Operations with these expressions required rules like the multiplication property of radicals and the distributive property. Practicing simplification and understanding how these rules play out will enhance one's skill in handling such algebraic expressions.