Problem 16 - Intermediate Algebra

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 Radical

Apply the distributive property to multiply \( \sqrt{3} \) with each term inside the parentheses. Start by multiplying \( \sqrt{3} \) by \( \sqrt{7} \):\[\sqrt{3} \cdot \sqrt{7} = \sqrt{3 \times 7} = \sqrt{21}.\]

Step 2: Distribute to the Second Term

Continue by distributing \( \sqrt{3} \) to \( \sqrt{10} \):\[\sqrt{3} \cdot \sqrt{10} = \sqrt{3 \times 10} = \sqrt{30}.\]

Step 3: Combine the Results

Combine the results from steps 1 and 2 to form the final expression:\[\sqrt{21} + \sqrt{30}.\]

Step 4: Simplify (if possible)

Check if \( \sqrt{21} \) or \( \sqrt{30} \) can be simplified further by finding any perfect square factors. Since neither \(21\) nor \(30\) have perfect square factors, the expression is already in its simplest radical form.

Key Concepts

Distributive Property

The distributive property is a useful algebraic property. It allows you to multiply a single term across a sum or difference within parentheses. - The general form is: If you have an expression of the form: \( a(b + c) \), it becomes \( ab + ac \). You distribute \( a \) to both \( b \) and \( c \) individually.When working with radicals, this principle remains the same. Consider the expression \( \sqrt{3}(\sqrt{7} + \sqrt{10}) \). - You distribute \( \sqrt{3} \) to \( \sqrt{7} \) to get \( \sqrt{3} \cdot \sqrt{7} = \sqrt{21} \).- Next, distribute \( \sqrt{3} \) to \( \sqrt{10} \) for \( \sqrt{3} \cdot \sqrt{10} = \sqrt{30} \).After distribution, simply combine terms to form the answer.

Radical Expressions

Radical expressions include square roots, cube roots, and other roots. A radical sign, \( \sqrt{} \), denotes the square root operation. - Square roots look for a number that multiplies by itself to make the number under the root.- Commonly used in algebra to indicate a radical number.When handling radical expressions:- Ensure each separate component is under one radical sign whenever multiplied.- Simplify by combining like terms if possible, mindful of coefficient and radical alike.For example, in solving \( \sqrt{21} + \sqrt{30} \), check both could have further simplification. However, no perfect squares multiply to 21 or 30, so they stay as is.

Multiplying Radicals

When multiplying radicals together, the product rule of radicals is your friend. It states: - To multiply \( \sqrt{a} \) and \( \sqrt{b} \), you write them as \( \sqrt{a \cdot b} \). - Always double-check if the multiplication results in a simpler form after multiplication.Let's apply this to our given problem:- Multiply \( \sqrt{3} \cdot \sqrt{7} \), leading to \( \sqrt{3 \cdot 7} = \sqrt{21} \).- Similarly, with \( \sqrt{3} \cdot \sqrt{10} \), see \( \sqrt{3 \cdot 10} = \sqrt{30} \).Finally, see if each can reduce further by identifying perfect squares. Both \( \sqrt{21} \) and \( \sqrt{30} \) remain, as no perfect square divides them. So, the expression is simplified in its radical form.