Problem 29 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt{2}(4 \sqrt{6}+2 \sqrt{7}) $$

Step-by-Step Solution

Answer

The simplified expression is \( 8\sqrt{3} + 2\sqrt{14} \).

Step 1: Distribute the Square Root

To solve this exercise, begin by distributing the factor \( \sqrt{2} \) to each term inside the parenthesis \( (4\sqrt{6} + 2\sqrt{7}) \). This means you will multiply \( \sqrt{2} \) with \( 4\sqrt{6} \) and \( 2\sqrt{7} \) separately.

Step 2: Multiply the First Terms

Multiply \( \sqrt{2} \) with the first term \( 4\sqrt{6} \):\[ \sqrt{2} \times 4\sqrt{6} = 4(\sqrt{2} \times \sqrt{6}) \].Using the property of radicals that \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \), you get: \[ 4 \sqrt{12} \].

Step 3: Simplify \( \sqrt{12} \)

Simplify \( \sqrt{12} \) by identifying perfect squares: \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \].Thus, \( 4 \sqrt{12} = 4 \times 2\sqrt{3} = 8\sqrt{3} \).

Step 4: Multiply the Second Terms

Multiply \( \sqrt{2} \) with the second term \( 2\sqrt{7} \): \[ \sqrt{2} \times 2\sqrt{7} = 2(\sqrt{2} \times \sqrt{7}) = 2\sqrt{14} \].

Step 5: Combine the Results

Combine the results from the multiplication to get the final expression:\[ 8\sqrt{3} + 2\sqrt{14} \].Since \( \sqrt{3} \) and \( \sqrt{14} \) are different, you cannot simplify further.

Key Concepts

Understanding Radicals

Radicals are mathematical symbols used to represent the root of a number. The most common radical is the square root, denoted by the symbol \( \sqrt{} \). When you see this symbol, it means you're looking for a number which, when multiplied by itself, gives the original number under the radical.
For example, the square root of 9 is 3, because \( 3 \times 3 = 9 \).

• Radicals can apply to other roots, not just square roots, like cube roots (\( \sqrt[3]{} \)).
• The radicand is the number under the radical symbol. In \( \sqrt{6} \), the number 6 is the radicand.
• Radicals have special rules: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).

This property is especially important when multiplying expressions that involve radicals, as it allows simplification through factorization.

Simplifying Expressions

Simplifying expressions involving radicals can often make them easier to understand and work with.
Here’s how you go about simplifying:

• Look for Perfect Squares: Check if there are any squares in the radicand. For \( \sqrt{12} \), notice that 12 can be expressed as \( 4 \times 3 \) where 4 is a perfect square.


• Break Down the Expression: Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). For \( \sqrt{12} \), it becomes \( \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \).


• Multiply Coefficients: Once simplified, multiply any coefficients outside the radicals together. If you have 4 in front of \( \sqrt{12} \), then simplify it to become \( 8\sqrt{3} \).

Simplifying can make solving the rest of the problem much more straightforward, especially when combined with other operations like addition or multiplication.

The Distributive Property with Radicals

The distributive property is a fundamental rule in algebra that lets you multiply a single term by each term inside a parenthesis.
This is crucial when dealing with expressions that contain sums or differences inside a parenthesis.

• This property is written as \( a(b + c) = ab + ac \).
• When you have radicals, apply this by multiplying outside numbers with each term inside the parenthesis. For example, \( \sqrt{2}(4\sqrt{6} + 2\sqrt{7}) \).
• By distributing \( \sqrt{2} \), it gets multiplied with each term: \( \sqrt{2} \times 4\sqrt{6} \) and \( \sqrt{2} \times 2\sqrt{7} \).


• This distribution helps to break down complex expressions into simpler parts, making them easier to handle.

Understanding this concept will unlock a lot of potential for solving more complicated algebraic expressions.