Problem 16 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(\sqrt{3}(\sqrt{7}+\sqrt{10})\)

Step-by-Step Solution

Answer

\( \sqrt{21} + \sqrt{30} \)

Step 1: Distribute the Square Root

The expression to solve is \( \sqrt{3}(\sqrt{7} + \sqrt{10}) \). Start by distributing \( \sqrt{3} \) to both terms inside the parentheses: \( \sqrt{3} \times \sqrt{7} + \sqrt{3} \times \sqrt{10} \).

Step 2: Simplify the Resulting Square Roots

Now, multiply the square roots one by one. \( \sqrt{3} \times \sqrt{7} = \sqrt{21} \), because \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). Similarly, \( \sqrt{3} \times \sqrt{10} = \sqrt{30} \).

Step 3: Combine the Terms

Combine the simplified terms to get the final expression: \( \sqrt{21} + \sqrt{30} \). Since there are no like terms or further simplifications available, this is the expression in its simplest form.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by a sum within parentheses. When we apply this property, we eliminate the parentheses by distributing the multiplication across each of the terms inside the parentheses. For example, with the expression \(a(b + c)\), the distributive property tells us to multiply \(a\) by both \(b\) and \(c\), resulting in the expression \(ab + ac\).

In our original exercise, we have \(\sqrt{3}(\sqrt{7} + \sqrt{10})\). By using the distributive property, we multiply \(\sqrt{3}\) by each element within the parentheses:
- \(\sqrt{3} \times \sqrt{7}\)- \(\sqrt{3} \times \sqrt{10}\)

This step-by-step expansion is crucial for simplifying complex algebraic expressions and is often the first step toward breaking down and solving problems involving multiplication over a sum.

Multiplying Square Roots

When multiplying square roots, an important rule to remember is \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\). This property allows us to combine the square roots by multiplying the numbers under the radical sign. This is a key process that makes handling radical expressions more straightforward.

In the exercise, after applying the distributive property, we individually multiply the square roots:

• \(\sqrt{3} \times \sqrt{7} = \sqrt{21}\)
• \(\sqrt{3} \times \sqrt{10} = \sqrt{30}\)

Each of these expressions can be simplified by multiplying the numbers under the roots, resulting in new square roots \(\sqrt{21}\) and \(\sqrt{30}\). This multiplication makes use of the property which helps in breaking down and simplifying expressions that include square roots.

Simplification in Algebra

Algebra often involves taking complex expressions and transforming them into simpler or more manageable forms. Simplification processes look to break down expressions to their most basic level without changing their value.

In the context of our problem, once the square root multiplications \(\sqrt{21} + \sqrt{30}\) have been performed, we check to see if further simplification is possible. In our case, each term under the square roots contain no perfect squares, and there are no like terms to combine. Hence, \(\sqrt{21} + \sqrt{30}\) is considered the simplest radical form.

Simplification ensures clarity and often serves to reveal the essence of the algebraic problem at hand, facilitating easier computation and interpretation. This process is frequently necessary in both solving equations and in making complex mathematical ideas more accessible and comprehensible.