Problem 11 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt{16} $$

Step-by-Step Solution

Answer

The square root of 16 is 4.

Step 1: Understanding the problem

We are tasked with evaluating the expression \( \sqrt{16} \), which means we need to find the square root of 16.

Step 2: Recall the definition of a square root

The square root of a number \( x \) is a value \( y \) such that \( y^2 = x \). In simpler terms, it's a number that, when multiplied by itself, results in the original number.

Step 3: Identify the square root

We need to identify which number, when squared (multiplied by itself), equals 16. Recall that for non-negative numbers, the square root is typically the non-negative root.

Step 4: Calculation

We know that \( 4 \times 4 = 16 \). Therefore, the square root of 16 is 4, since \( 4^2 = 16 \).

Step 5: Solution

Since no other non-negative number squared equals 16, the primary square root of 16 is 4. Hence, \( \sqrt{16} = 4 \).

Key Concepts

Real Numbers

Real numbers are an essential concept in mathematics and include all the numbers you can think of. They encompass a wide variety of numbers such as whole numbers, decimals, fractions, and irrational numbers.

Here’s a quick breakdown of what real numbers include:

• Natural Numbers: These are counting numbers like 1, 2, 3, and so on.
• Whole Numbers: This group includes all natural numbers plus zero. So, 0, 1, 2, 3, etc.
• Integers: These are whole numbers and their negative counterparts like -2, -1, 0, 1, 2.
• Rational Numbers: Numbers that can be expressed as a fraction of two integers, like \( \frac{1}{2} \) or 0.75.
• Irrational Numbers: Numbers that cannot be neatly expressed as a simple fraction, like \( \pi \) or the square root of 2.

This diversity allows real numbers to form an unbroken number line that includes every possible magnitude and direction. When dealing with concepts like square roots, having the real number set means you often work with positive results when asked for square roots.

Square Numbers

Square numbers or perfect squares are the results you get when multiplying a number by itself. In other words, when you take any whole number \( n \) and compute \( n \times n \), you get a square number.

Some common examples include:

• 1, because \(1 \times 1 = 1\)
• 4, because \(2 \times 2 = 4\)
• 9, because \(3 \times 3 = 9\)
• 16, because \(4 \times 4 = 16\)
• 25, because \(5 \times 5 = 25\)

Square numbers are helpful because they make recognizing the square roots easier. You simply reverse the operation to find the original number that was squared. In the case of our exercise, you can see that because 16 is a perfect square, it's a straightforward process to find that its square root is 4.

Definition of Square Root

The concept of a square root is foundational in mathematics. A square root of a number is a value that, when multiplied by itself, returns the original number. You can think of it as unlocking the number that was hidden through the multiplication process.

For example, when we say \( \sqrt{16} \), we are looking for the number that, when squared (or multiplied by itself), results in 16. Here, that number is 4 because \( 4 \times 4 = 16 \).
Another important detail is that square roots are usually requested in their principal or non-negative form, which means we typically provide the positive root unless specified otherwise.

• Example of square root operation: \( \sqrt{25} = 5 \) since \(5 \times 5 = 25\).
• Negative roots also exist but are typically noted differently or in special contexts.

Understanding square roots is critical since they appear in various mathematical contexts, from solving quadratic equations to analyzing geometrical shapes.