Problem 6 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

4

Step 1: Simplify inner square root

First, we need to simplify the expression inside the inner square root, \( \sqrt{256} \). Calculate the square root of 256. Since 256 is a perfect square, its square root is 16. Thus, \( \sqrt{256} = 16 \).

Step 2: Simplify outer square root

Now that we have simplified the inner square root, we have the expression \( \sqrt{16} \) to calculate. Since 16 is also a perfect square, its square root is 4. Hence, \( \sqrt{16} = 4 \).

Step 3: Conclusion

After calculating both square roots, the simplified expression of \( \sqrt{\sqrt{256}} \) is 4.

Key Concepts

Square Roots

Square roots are mathematical operations used to find a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 multiplied by itself equals 9. This operation is symbolized by a radical sign, \( \sqrt{} \). It is essential for students to recognize that not every number has a neat square root, such as 2 or 3, which result in irrational numbers.
The process often involves estimation or rationalization, but in this case, with \( \sqrt{\sqrt{256}} \), the task is more straightforward because the numbers are perfect squares.

Perfect Squares

Perfect squares are numbers that result from squaring an integer. Examples include 1, 4, 9, 16, 25, and so on. Each of these can be written as \( n^2 \), where \( n \) is an integer. In algebraic simplification, identifying these numbers simplifies the calculations significantly.
In our exercise, recognizing that 256 is a perfect square helps us know that \( \sqrt{256} = 16 \). Similarly, because 16 is also a perfect square, we recognize that \( \sqrt{16} = 4 \). Utilizing perfect squares allows us to quickly and efficiently find square roots without extensive calculation.

Step-by-Step Solutions

Breaking problems into smaller, manageable steps is crucial for understanding complex expressions. In the given exercise \( \sqrt{\sqrt{256}} \), the problem was decomposed into two phases: simplifying the inner square root and then the outer square root.

• **Step 1:** Simplify the inner expression \( \sqrt{256} \). Since 256 is a perfect square, its square root is easily found to be 16.
• **Step 2:** Simplify the resulting expression \( \sqrt{16} \). Again, because 16 is also a perfect square, the square root calculation is straightforward, yielding 4.
• **Conclusion:** After individually finding the square roots, the final simplified expression is 4.

This approach not only makes solving simpler but also enhances conceptual understanding, benefiting the learner both in this specific problem and in broader mathematical challenges.