Problem 6 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{\sqrt{256}} $$

Step-by-Step Solution

Answer

4

Step 1: Understand the Expression

The given expression is \( \sqrt{\sqrt{256}} \), which means we have a nested square root that needs to be simplified.

Step 2: Simplify the Inner Square Root

First, simplify the inner expression \( \sqrt{256} \). We know that 256 is a perfect square, and \( 256 = 16^2 \). So, \( \sqrt{256} = 16 \).

Step 3: Simplify the Outer Square Root

Now that we have simplified the inner square root to 16, the expression becomes \( \sqrt{16} \). Since \( 16 = 4^2 \), we know \( \sqrt{16} = 4 \).

Key Concepts

Understanding Nested Square Roots

A nested square root is essentially a square root within a square root, creating a layered expression of roots. This can make simplification seem a bit challenging at first. To tackle this, always work from the innermost root outwards. Thankfully, by solving each inner square root step-by-step, the complexity diminishes.

• Start with the innermost square root and simplify it first.
• Gradually work your way to the outer square roots.
• This systematic approach ensures a simpler final expression.

In the given exercise, we have the expression \( \sqrt{\sqrt{256}} \). By simplifying the inner \( \sqrt{256} \) first, then addressing the resulting \( \sqrt{16} \), we methodically simplify the nested roots. This approach is crucial for tackling similar complex problems involving nested radicals.

The Role of Perfect Squares

Perfect squares play a vital role when it comes to simplifying square roots. A perfect square is a number that has an integer as its square root. Recognizing these numbers is essential because they make simplifying radical expressions straightforward.
Some examples of perfect squares include:

• 4 (since \( 2^2 = 4 \))
• 9 (since \( 3^2 = 9 \))
• 16 (since \( 4^2 = 16 \))
• 256 (since \( 16^2 = 256 \))

Knowing these can save time and effort when simplifying expressions. In the problem at hand, recognizing that both 256 and 16 are perfect squares allowed us to easily reduce \( \sqrt{\sqrt{256}} \) to a simple integer 4. This is why familiarizing yourself with common perfect squares is incredibly beneficial.

Demystifying Radical Expressions

Radical expressions often involve roots like square roots and can seem complicated at first. However, with practice, they become much easier to handle. A radical expression is essentially any expression that contains a root. Here’s how to approach them effectively:

• Identify if the number under the root is a perfect square. If so, take its square root easily.
• Break down complex expressions into simpler parts, such as dealing with one root at a time for nested radicals.
• Use factoring if necessary to find equivalent expressions that are easier to work with.

By following these strategies, you can simplify many radical expressions, even those that seem complicated at first glance. For our featured exercise, applying these principles allowed us to simplify from an initially daunting \( \sqrt{\sqrt{256}} \) to just 4. As you increase your familiarity with radical expressions, you'll find tackling them becomes much more intuitive.