Problem 18 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt{1.44} $$

Step-by-Step Solution

Answer

The value of \( \sqrt{1.44} \) is 1.2.

Step 1: Understand the Expression

You need to evaluate the square root of 1.44, which is written as \(\sqrt{1.44}\). When you apply the square root, you are looking for a number that, when multiplied by itself, gives 1.44.

Step 2: Simplify the Square Root

Try to find factors of 1.44 that break down into perfect squares. Notice that 1.44 can be written as \( (1.2)^2 \) since \( 1.2 \times 1.2 = 1.44 \).

Step 3: Evaluate the Square Root

Now, use the simplification from Step 2: since \( 1.44 = (1.2)^2 \), it follows that \( \sqrt{1.44} = 1.2 \), because the square root cancels the square.

Key Concepts

Real Numbers

Real numbers encompass a vast set of numbers that include all rational and irrational numbers. They can be found on the number line, extending infinitely in both the positive and negative directions. Real numbers are incredibly important in mathematics because they allow for precise representation of values in everyday calculations, which is why they are used to evaluate expressions.

Here's a simple breakdown of the types of real numbers:

• Rational Numbers: These numbers can be written as fractions or ratios of integers, such as \( \frac{1}{2} \) or 0.75.
• Irrational Numbers: Numbers that cannot be precisely written as fractions, like \( \pi \) or \( \sqrt{2} \). They have non-repeating, non-terminating decimal expansions.

Understanding where square roots fall into this spectrum helps in calculating and simplifying them, as seen in the evaluation of \( \sqrt{1.44} \). This is an example of dealing with real numbers, demonstrating how both rational and irrational elements work within the system.

Simplifying Radicals

Simplifying radicals involves breaking down a root expression to its simplest form. This process often includes finding factors that are perfect squares to make computations easier.

When we simplify \( \sqrt{1.44} \), the first step is to determine whether 1.44 can be expressed as a perfect square. This would allow us to redefine the square root to remove the radical expression. Simplification steps include:

• Identifying the perfect square factor, here 1.44 equals \( (1.2)^2 \).
• Applying the knowledge that the square and the square root are inverse operations, leading to \( \sqrt{(1.2)^2} = 1.2 \).

Recognizing such patterns not only makes simplification quicker but also helps in understanding the relationship between numbers and their root expressions. Making radicals easier to work with is key in both basic and advanced mathematics.

Perfect Squares

Perfect squares are special numbers that arise when an integer is multiplied by itself. Recognizing these numbers allows for easier simplification of square roots and algebraic expressions. In our example, understanding that 1.44 is a perfect square simplified our task significantly.

Important points about perfect squares:

• They always result from squaring either an integer or a decimal.
• Examples include 1, 4, 9, 16, and for decimals, 2.56 (since \( 1.6 \times 1.6 = 2.56 \)).
• Identifying a number as a perfect square instantly tells us its square root without further calculation. For instance, knowing that \( (1.2)^2 = 1.44 \) allows us to confidently state \( \sqrt{1.44} = 1.2 \).

This understanding underlines the importance of recognizing perfect squares in simplifying and evaluating mathematical expressions. It forms the basis for not just square roots, but also more complex algebraic manipulations.