Problem 2 - College Algebra

Question

Evaluate each expression in Exercises \(1-6\) or indicate that the root is not a real number. $$\sqrt{25}$$

Step-by-Step Solution

Answer

\(\sqrt{25} = 5\)

Step 1: Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, results in the original number. Hence, the square root of 25 can be found by finding a number that, when multiplied by itself, equals to 25.

Step 2: Calculate the Square Root

Calculate the square root of 25. This can be done by considering the numbers which when multiplied by itself equals to 25. Thus, the number 5 when multiplied by itself (i.e., \(5*5\) or \(5^2\)) equals to 25.

Step 3: Expression Evaluation

Hence, the value of the expression \(\sqrt{25}\) is confirmed as 5, since \(5*5 = 25\).

Key Concepts

Evaluate Square Roots

When we talk about evaluating square roots, we're looking for the number that produces a specific value when multiplied by itself. For students starting out, it can seem tricky, but it's really just about finding pairs of identical factors.
Take the square root of 9, for example. The question you're asking is 'What number times itself equals 9?' The answer, as you may know, is 3 because of the multiplication fact that 3 times 3 equals 9 (\(3^2 = 9\)). By the same logic, when we evaluate the square root of 25, we're looking for a number that, when multiplied by itself, gives us 25.

Radical Expressions

Radical expressions are mathematical phrases that include a radical symbol (\(\sqrt{}\)) with a number inside it. The most common type of radical expression is a square root. The expression \(\sqrt{x}\), for instance, represents the positive square root of x. It's important to highlight here that square roots have two answers: a positive and a negative root (±). However, when we see a radical symbol without any additional notation, it implies the principal (or positive) root.
That means whenever you see \(\sqrt{25}\), you're only dealing with the positive square root, which in this case is 5. Remember also that not all numbers have real square roots. The square root of a negative number, by standard real number operations, is not a real number but rather an imaginary one.

Real Numbers

Real numbers include all the numbers on the number line. This vast category contains the whole numbers, fractions, decimals, and irrationals like \(\sqrt{2}\) or π. When we talk about the square roots being real, we mean that you can find them on the number line.
A key point to remember is that the square roots of positive numbers are always real numbers. This explanation becomes particularly crucial when dealing with exercises such as evaluating the square root of 25. As 25 is a positive whole number, its square root is also a real number. The tricky part comes in when evaluating the square root of a negative number, which falls outside the scope of real numbers and instead belongs to the imaginary number category.

Square Root of 25

To tackle the square root of 25 head-on, you simply need to find the number that, when squared (multiplied by itself), equals 25. You might already remember this number from your multiplication tables — it's 5.
The calculation would look like this: if \(x = \sqrt{25}\), then to find x, you solve the equation \(x^2 = 25\). The solution, \(x = 5\), is straightforward because 5 is a commonly known perfect square. It's also worth noting that 25 is a perfect square, meaning it derives from squaring an integer.
In other subjects, such as geometry or physics, knowing these perfect squares and their roots can be essential for solving more complex problems involving areas or other square-related formulas.