Problem 45 - Elementary Algebra

Question

For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number." $$ \sqrt{64} $$

Step-by-Step Solution

Answer

Answer: The square root of 64 is 8.

Step 1: Identify the number given

The number given in the exercise is 64, and we need to find its square root.

Step 2: Recall the definition of square root

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, if \(a \times a = b\), then \(\sqrt{b} = a\).

Step 3: Find the square root of 64

Now we need to find the square root of 64. We know that \(\sqrt{64}\) can be written as either \(-8 \times -8 = 64\) or \(8 \times 8 = 64\). Therefore, the square root of 64 is 8. The final result is: $$\sqrt{64} = 8$$

Key Concepts

Square Root Definition

Understanding the concept of a square root is essential when delving into basic algebra. A square root is essentially a number that produces a specified quantity when it's multiplied by itself. To put it in simple terms, if you multiply the square root of a number by itself, you'll get the original number back. For instance, if you consider the equation \( a \times a = b \), the square root of \( b \) is represented by \( a \). This is denoted in mathematics as \( \sqrt{b} = a \).

When evaluating square roots, it's crucial to remember that there can be both positive and negative roots because both \( (-a) \times (-a) \) and \( a \times a \) will yield \( b \). However, when we talk about 'the square root' in most contexts, we are referring to the positive square root, known in mathematical terms as the 'principal square root'.

Perfect Squares

A perfect square is a number that is the square of an integer. In other words, it's the product of some integer with itself. For example, numbers like 1, 4, 9, 16, 25, and so on are perfect squares because they can be expressed as \( 1^2 \), \( 2^2 \), \( 3^2 \), \( 4^2 \) and \( 5^2 \) respectively.

Familiarizing yourself with the list of perfect squares can make the process of evaluating square roots much faster and easier. When you encounter a square root of a perfect square in an exercise, it simplifies directly to a known number without the need for approximation or further calculation. An example is \( \sqrt{64} \), which we know is 8 because 8 is the number that multiplied by itself gives 64. It's also useful to note that non-perfect squares will result in irrational numbers, which can't be expressed precisely as simple fractions.

Real Numbers

Real numbers are the set of numbers that include both rational and irrational numbers, and they can be found on the number line. Rational numbers are those that can be expressed as a fraction with both the numerator and denominator as integers, whereas irrational numbers cannot be fully written out as a fraction and have non-repeating, non-terminating decimal parts.

In the context of square roots, they can either be rational or irrational numbers. For instance, the square root of a perfect square, like 64, is a rational number (8 in this case). However, if we take a non-perfect square, like 2, the square root is an irrational number represented approximately as 1.414. It's also important to remember that the square roots of negative numbers are not considered real numbers, instead they are referred to as imaginary numbers because they can't be plotted on the traditional number line.