Problem 16 - Algebra 2

Question

Find each real-number root. $$ \sqrt{0.36} $$

Step-by-Step Solution

Answer

The real-number root of \(\sqrt{0.36}\) is 0.6.

Step 1: Identify the radicand

The number under the square root sign is known as the radicand. Here, the radicand is 0.36.

Step 2: Find the principal square root

To find the principal (positive) square root of a positive real number, we look for a number that, when multiplied by itself, gives the radicand. The square root of 0.36 is the number that, when squared, results in 0.36.

Step 3: Determine the Positive Real-Number Root

Since 0.6 times 0.6 equals 0.36, the principal square root of 0.36 is 0.6.

Key Concepts

Square Root

Understanding the square root of a number is fundamental in mathematics. It is represented by the symbol \( \sqrt{} \) and refers to finding a value that, when multiplied by itself, gives the original number. For example, to calculate the square root of 16, you would look for a number that, when squared, equals 16. The answer is 4 since \( 4 \times 4 = 16 \).

When computing the square root, there can actually be two roots: one positive and one negative. Thus, the number 4 has two square roots: \( +4 \) and \( -4 \) because \( (+4) \times (+4) = 16 \) and also \( (-4) \times (-4) = 16 \). However, when we talk about the square root, we typically refer to the principal square root, which is the positive root.

Radicand

The term radicand refers to the number under the square root sign, waiting to be 'liberated' by the operation. In the exercise \( \sqrt{0.36} \) the radicand is 0.36. It is important to recognize the radicand because it is the target of our calculation, the mystery box's content we eagerly want to reveal.

In more complex expressions, the radicand can be a whole expression itself. Therefore, when faced with such tasks, separating and identifying the radicand is a crucial starting point for simplification. Always ensure the radicand is a non-negative number because square roots of negative numbers lead us into the realm of imaginary numbers, which requires a slightly different approach.

Principal Square Root

The principal square root is the non-negative root of a number and is what most people refer to when they speak of the square root. For the example given, the square root of 0.36, the principal square root is 0.6. That's because \(0.6 \times 0.6 = 0.36\).

In contrast to the principal root, the negative root is often overlooked because it is not commonly known as 'the square root.' Nevertheless, it's crucial in certain contexts, particularly in higher mathematics, to include both roots. When expressing both roots, we would write \( \pm 0.6 \), which denotes both \( +0.6 \) and \( -0.6 \) as the roots of 0.36.