Problem 122 - Elementary and Intermediate Algebra

Question

a. \(\sqrt{16}\) b. \(\sqrt[4]{16}\)

Step-by-Step Solution

Answer

a. \( \sqrt{16} = 4 \); b. \( \sqrt[4]{16} = 2 \).

Step 1: Identify the square root

The square root of a number is a value that, when multiplied by itself, gives the original number. We need to find a number that, when squared, equals 16.

Step 2: Calculate the square root

For part a, find the square root of 16. Since 4 times 4 equals 16, the square root of 16 is 4, i.e., \( \sqrt{16} = 4 \).

Step 3: Identify the fourth root

The fourth root of a number is a value that, when raised to the fourth power, gives the original number. We need to find a number that, when raised to the fourth power, equals 16.

Step 4: Calculate the fourth root

For part b, find the fourth root of 16. Since 2 raised to the power of 4 equals 16, the fourth root of 16 is 2, i.e., \( \sqrt[4]{16} = 2 \).

Key Concepts

Understanding the Square Root

The square root is a mathematical concept that finds a number which, when multiplied by itself, results in the original given number. For instance, if we consider the number 16, to find its square root, we are looking for a number which, when squared, equals 16. We can think of this process as reverse multiplication, where we identify a perfect match that makes the equation true. In the case of \( \sqrt{16} \), the number 4 is the correct answer because \( 4 \times 4 = 16 \). This discovery process essentially reimagines multiplication as a way to split a number evenly into its two root components.

Here are a few tips to calculate square roots more effectively:

• Recognize perfect squares like 1, 4, 9, 16, 25, and so on. This helps identify numbers that are easily square-rooted.
• Break down complex numbers into simpler factors to find their square roots.
• Practice makes perfect—using square root tables or calculators can help visualize this concept better in practice.

By embracing these strategies, understanding square roots will become more intuitive.

Demystifying the Fourth Root

Just as the square root focuses on finding a number that needs to be multiplied once by itself, the fourth root involves finding a number which, when multiplied by itself four times, returns the original number. In simple terms, when dealing with \( \sqrt[4]{16} \), we need to pinpoint a number 'x' such that \( x^4 = 16 \). This sounds more complex but works similarly to finding square roots.

• For example, 2 is the fourth root of 16 because \( 2 \times 2 \times 2 \times 2 = 16 \), which means \( 2^4 = 16 \).
• Fourth roots tend to be utilized when dealing with exponents or when simplifying radical expressions.

Breaking down complex problems step-by-step and simplifying the root extraction process can make these calculations less intimidating. Clarity on these concepts is essential, especially in advanced math topics.

Step-by-Step Solutions

Step-by-step solutions help demystify math problems by breaking them into manageable parts. Addressing mathematical problems in this way means tackling each element separately before moving on to the next.

Here is a simple strategy used in the previous exercise:

• **Step 1:** Recognize what kind of root problem it is - in this exercise, parts (a) and (b) addressed both square and fourth roots.
• **Step 2:** Use your knowledge on root calculations to find the answer for each case.
  For \( \sqrt{16} \), identify the number that, when squared, equals 16. Similarly, for \( \sqrt[4]{16} \), find the number that raised to the fourth power equals 16.
• **Step 3:** Compute and verify the roots, ensuring that your calculated number accurately aligns with the multiplication to return to the original number.

By persistently applying step-by-step solutions, you not only solidify your understanding but boost your confidence in solving mathematical challenges.