Problem 38 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \sqrt{3}(\sqrt{5}-3) $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(\sqrt{3}(\sqrt{5}-3)\) Answer: \(\sqrt{15} - 3\sqrt{3}\)

Step 1: Distributive Property of Multiplication

We will use the distributive property of multiplication, which states that \(a(b+c) = ab + ac\). In our case, \(a = \sqrt{3}\), \(b = \sqrt{5}\), and \(c = -3\).

Step 2: Distribute \(\sqrt{3}\)

We will now distribute the \(\sqrt{3}\) to both terms inside the parentheses: $$\sqrt{3}(\sqrt{5}-3) = \sqrt{3} \cdot \sqrt{5} + \sqrt{3} \cdot (-3)$$

Step 3: Multiply the Terms

Let's now multiply the terms: $$\sqrt{3} \cdot \sqrt{5} = \sqrt{15}$$ and $$\sqrt{3} \cdot (-3) = -3\sqrt{3}$$

Step 4: Combine the Results

Finally, we combine the results from the previous step: $$\sqrt{15} - 3\sqrt{3}$$ The simplified expression is \(\sqrt{15} - 3\sqrt{3}\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in math, especially in algebra. It allows you to simplify expressions by multiplying each term inside a set of parentheses by a factor outside of it. This property can be written as:

• For any numbers or expressions, a, b, and c, the distributive property is expressed as: \(a(b + c) = ab + ac\).

In the given exercise, we applied the distributive property to the expression \(\sqrt{3}(\sqrt{5} - 3)\). Here, \(\sqrt{3}\) is multiplied by both terms inside the parentheses—first by \(\sqrt{5}\) and then by \(-3\). This helps us break down the problem into simpler parts that are much easier to handle. This approach of using the distributive property is very useful, not only when working with radical expressions but also in various algebraic expressions you will encounter.

Square Roots in Algebra

Square roots are a type of radical expression and are common in algebra. A square root asks "what number, when multiplied by itself, gives the original number?" The square root of \(n\) is indicated by \(\sqrt{n}\).

• Some important properties include \(\sqrt{a} \times \sqrt{a} = a\), and \(\sqrt{a \cdot b} = \sqrt{a} \times \sqrt{b}\).

In algebraic simplification, it's important to recognize square roots and understand how they can be manipulated. For instance, \(\sqrt{15}\) results from multiplying \(\sqrt{3}\) and \(\sqrt{5}\). Knowing these rules helps you manage expressions that include square roots effectively, aiding in both simplifying expressions and solving equations that involve radical terms.

Multiplication of Radicals

The multiplication of radicals involves multiplying numbers under the square root sign (or other roots).

• When multiplying \(\sqrt{a}\) and \(\sqrt{b}\), you get \(\sqrt{a \cdot b}\).

In the exercise, we multiplied the terms \(\sqrt{3}\) and \(\sqrt{5}\) which simplifies directly to \(\sqrt{15}\). Also, when multiplying a radical with a number like \(-3\), such as \(\sqrt{3} \cdot (-3)\), it results in \(-3\sqrt{3}\). This method of multiplying radicals directly underlines the role of the distributive property and confirms that operations between radicals can often maintain their radical characteristics, ensuring expressions remain simplified and manageable.