Problem 32 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{2}(\sqrt{8}-4) $$

Step-by-Step Solution

Answer

The simplified expression is \( 4 - 4\sqrt{2} \)

Step 1: Simplify the Radical

Simplify the radical \( \sqrt{8} \). To do this, find two factors of 8 where one of the factors is a perfect square. We can take 4 and 2. So, \( \sqrt{8} \) can be written as \( \sqrt{4} \times \sqrt{2} \), which simplifies to \( 2\sqrt{2} \). Now the expression looks like: \( \sqrt{2}(2\sqrt{2}-4) \)

Step 2: Perform the Product

Perform the multiplication operation in the brackets. Multiplying \( \sqrt{2} \) by \( 2\sqrt{2} \) gives 4 and \( \sqrt{2} \) multiplied by -4 gives \( -4\sqrt{2} \). Thus, the expression simplifies to: \( 4 - 4\sqrt{2} \)

Key Concepts

Radical Expressions

Radical expressions involve numbers or variables under a radical sign, commonly square roots. Understanding these expressions is crucial for simplifying complex equations. Radicals can often make math problems look more challenging than they really are. However, by learning to simplify these radicals, you can make the math more manageable.

Let's say we have a square root, such as \( \sqrt{8} \). This can be expressed in terms of its factors. Identifying which numbers are perfect squares can help to simplify it. In this example, realizing 8 is made up of the factors 4 and 2 is key. From here, we can write \( \sqrt{8} \) as \( \sqrt{4} \times \sqrt{2} \). This brings us to simplify it further as \( 2\sqrt{2} \). Now, our radical expression is much simpler.

Perfect Squares

Perfect squares are numbers which can be expressed as the product of an integer with itself. For example, 4 is a perfect square because it is \( 2 \times 2 \). Recognizing perfect squares is valuable when simplifying radicals.

In the expression \( \sqrt{8} \), identifying 4 as a perfect square allows us to break it down into \( \sqrt{4} \times \sqrt{2} \). Since \( \sqrt{4} \) equals 2, we are left with \( 2\sqrt{2} \). Knowing perfect squares simplifies radical expressions significantly and streamlines the simplification process, saving time and reducing potential mistakes.

Multiplication of Radicals

Multiplying radicals might seem tricky, but it's straightforward once you understand the basics. To multiply radicals, multiply the numbers outside the radicals separately from those inside. If you have \( \sqrt{a} \times \sqrt{b} \), it becomes \( \sqrt{ab} \).

For example, consider \( \sqrt{2}(2\sqrt{2}) \). First, multiply the numbers outside the radicals: \( 2 \times 1 = 2 \). Then, inside the radicals: \( \sqrt{2} \times \sqrt{2} = \sqrt{4} \), which is 2. Together, this gives us 4. Similarly, when you multiply \( \sqrt{2} \) by -4, you handle the number -4 separately and then multiply it by \( \sqrt{2} \). This rules results in \( -4\sqrt{2} \). Thus, multiplying radicals involves systematic approaches that make complex calculations simpler.