Problem 41 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \sqrt{8}(\sqrt{3}+\sqrt{2}) $$

Step-by-Step Solution

Answer

Question: Simplify the expression: √8(√3+√2) Answer: 2√6+4

Step 1: Write Down the Expression

Write down the given expression. $$ \sqrt{8}(\sqrt{3}+\sqrt{2}) $$

Step 2: Apply the Distributive Property

Multiply every term inside the parentheses by the square root of 8. $$ (\sqrt{8} \cdot \sqrt{3}) + (\sqrt{8} \cdot \sqrt{2}) $$

Step 3: Simplify the Inside Terms

Simplify the multiplication of the square roots using the properties of radicals. $$ \sqrt{8 \cdot 3} + \sqrt{8 \cdot 2} $$

Step 4: Calculate the Products

Calculate the products inside the square roots. $$ \sqrt{24}+\sqrt{16} $$

Step 5: Simplify the Radicals

Recognize that 24 can be factored as 4 times 6, and 16 is a perfect square. Simplify the radicals further. $$ \sqrt{4 \cdot 6}+\sqrt{16} $$ $$ 2\sqrt{6}+4 $$

Step 6: Write the Final Answer

The simplified expression is: $$ 2\sqrt{6}+4 $$

Key Concepts

Distributive Property

The distributive property is a fundamental math principle that allows you to distribute a single term over a sum or difference within parentheses. This means you multiply the term on the outside with every term inside the parentheses. For example, in the expression \( a(b+c) \), applying the distributive property gives us \( ab + ac \). It splits the expression into separate terms that can be solved or further simplified.

In the context of radical expressions, like in our original exercise \( \sqrt{8}(\sqrt{3} + \sqrt{2}) \), we use the distributive property to multiply \( \sqrt{8} \) with both \( \sqrt{3} \) and \( \sqrt{2} \).

• This operation transforms the expression into \( (\sqrt{8} \cdot \sqrt{3}) + (\sqrt{8} \cdot \sqrt{2}) \).

This step sets the stage for further simplification by applying other properties of radicals.

Properties of Radicals

Understanding the properties of radicals helps in simplifying expressions that involve square roots. A few key properties make this process easier, particularly when you're performing multiplication or looking to simplify expressions.

• The product of square roots: \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \).
• Similarly, \( \sqrt{a^2} = a \), which is fundamental when working with perfect squares inside the radical to simplify expressions efficiently.

These properties allow us to manipulate and break down expressions, as demonstrated in the original problem. After applying the distributive property, we calculated \( \sqrt{8 \cdot 3} + \sqrt{8 \cdot 2} \), examining the expression with these radical properties to simplify further.

As you observe, multiplying the interiors and recognizing the square root of any perfect squares, such as turning \( \sqrt{16} \) into 4, leads to the simplification of the whole expression.

Simplifying Radicals

Simplifying radicals revolves around making expressions easier to manage by reducing the numbers under the radical sign to their simplest form. This often involves identifying perfect squares.

For a number like \( \sqrt{24} \), we see 24 can be broken into \( 4 \times 6 \). Since 4 is a perfect square, we can simplify further by determining that \( \sqrt{4} \) is 2. Thus, \( \sqrt{24} \) becomes \( 2\sqrt{6} \).

In the problem presented, simplifying steps led us from \( \sqrt{24} + \sqrt{16} \) to \( 2\sqrt{6} + 4 \).

• This involves recognizing that \( \sqrt{16} \) simplifies directly to 4 as it is perfect.
• Then, simplifying \( \sqrt{24} \) to \( 2\sqrt{6} \), utilizing properties of square roots solidifies the final expression.

This methodical approach ensures that every radical is expressed in its simplest form, making further calculations or understanding of the expression straightforward.