Problem 69 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{2}(\sqrt{8}+\sqrt{6}) $$

Step-by-Step Solution

Answer

Answer: The product of the given expression is \(4 + \sqrt{12}\).

Step 1: Simplify the terms in parentheses

Write the expression with simplified square roots: $$ \sqrt{2} (\sqrt{4 \cdot 2} + \sqrt{2 \cdot 3}) $$

Step 2: Simplify the square roots

Now we will simplify the square roots in the parentheses: $$ \sqrt{2} (2\sqrt{2} + \sqrt{6}) $$

Step 3: Apply the distributive property

Now we will distribute the \(\sqrt{2}\) term to both terms inside the parentheses: $$ (\sqrt{2} \cdot 2\sqrt{2}) + (\sqrt{2} \cdot \sqrt{6}) $$

Step 4: Calculate the products

Calculate the products in the expression: $$ (2 \cdot 2) + (\sqrt{2 \cdot 6}) $$

Step 5: Simplify the final expression

Finally, we simplify the expression to obtain the final result: $$ 4 + \sqrt{12} $$ The product of the given expression is \(4 + \sqrt{12}\).

Key Concepts

Simplifying Square Roots

Simplifying square roots is a useful skill in mathematics that helps make expressions easier to handle. The idea is to break down the square root into simpler parts that we can easily identify and handle. For example, consider the square root of 8:

• First, we look for a perfect square factor of 8, which is 4. So, we can rewrite \( \sqrt{8} \) as \( \sqrt{4 \times 2} \).
• Next, we separate this into two roots: \( \sqrt{4} \) and \( \sqrt{2} \).
• We know that \( \sqrt{4} = 2 \), so this simplifies to \( 2 \sqrt{2} \).

Breaking down radicals to their simplest form not only makes calculations easier but often uncovers patterns that are helpful in solving algebraic problems. Always look for those perfect square factors to simplify your work when dealing with square roots.

Distributive Property

The distributive property is an essential principle in mathematics, especially when dealing with expressions involving addition and multiplication. It states that for any numbers or expressions \( a \), \( b \), and \( c \), \( a(b + c) = ab + ac \).

• This property allows us to multiply a single term across terms inside a set of parentheses.
• In the context of our example, \( \sqrt{2} (2\sqrt{2} + \sqrt{6}) \), the distributive property lets us multiply \( \sqrt{2} \) by each term inside the parentheses individually.

Breaking it down:

• \( \sqrt{2} \cdot 2\sqrt{2} \): multiply the constants first ( \( \sqrt{2} \cdot 2 = 2\sqrt{2} \)). When squared, this component becomes \( 4 \).
• \( \sqrt{2} \cdot \sqrt{6} \): remember the property of multiplying square roots, \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \). Hence, \( \sqrt{2 \cdot 6} \).

The distributive property is key in rearranging and simplifying expressions, making them easier to manage and understand in algebra.

Algebraic Expressions

Algebraic expressions are combinations of symbols and numbers that represent mathematical operations. They can be as simple as a single number or as complex as a multi-variable equation. Key characteristics include:

• Terms: The individual parts of an expression separated by plus or minus signs (e.g., \( 2x + 3 \) has two terms).
• Coefficients: Numbers that multiply the variables (in \( 3x \), 3 is the coefficient).
• Variables: Symbols like \( x \), \( y \), and \( z \) that are placeholders for numbers.
• Constants: Numbers without variables (e.g., \( 5 \) in \( x + 5 \)).

In the problem \( \sqrt{2}(\sqrt{8} + \sqrt{6}) \), each term and operation must be considered carefully:

• Breakdown of components: We have both radical terms (e.g., \( \sqrt{8} \)) that need to be simplified and operations involving the distributive property.
• Combination of simplified terms: By careful simplification and multiplication, we combine terms into an understandable form like \( 4 + \sqrt{12} \).

Understanding and manipulating algebraic expressions is pivotal in subsequently solving equations and finding solutions to practical problems.