Problem 56 - Algebra 2

Question

Simplify each radical expression. \(\sqrt{144}\)

Step-by-Step Solution

Answer

The simplified expression is 12.

Step 1: Identify Perfect Square

Examine the number under the square root symbol. We need to determine if any perfect squares can be found. Since 144 is a perfect square and equals \(12^2\), identify 144 as a perfect square.

Step 2: Simplify the Square Root

Find the square root of 144. The square root of a perfect square is simply the integer that, when multiplied by itself, results in the number under the square root. Therefore, \( \sqrt{144} = 12 \) because \(12 \times 12 = 144\).

Key Concepts

Understanding Radical Expressions

Radical expressions are mathematical terms that include a radical symbol, like a square root. They represent the root of a number or expression. In simple terms, they help us find a number which, when used a certain number of times in a multiplication, results in the given number. The most common radical is the square root, symbolized by \( \sqrt{} \) which takes the square root of the number under it. Other radicals include cube roots and fourth roots, but square roots are used the most.

When simplifying radical expressions, the goal is to make them as simple as possible while maintaining equality. It helps in reducing complex calculations and making numbers more manageable for further mathematical operations. For example, when simplifying \( \sqrt{144} \), the result is 12. Simplifications often involve finding and using perfect squares because they have exact integer values as square roots.

The Role of Perfect Squares

Perfect squares are numbers that have whole numbers as their square roots. These are numbers like 1, 4, 9, 16, 25, and so on, where each number can be expressed as an integer squared. For instance, \( 144 = 12^2 \), meaning that 144 is a perfect square and its square root is 12.

Identifying perfect squares is a crucial step in simplifying radical expressions. It allows us to replace a complex square root with a simpler whole number. When you know that 144 is a perfect square, you can quickly find its square root without complex calculations. Knowing common perfect squares by memory is helpful because it speeds up the simplification process.

Grasping Square Roots

A square root answers the question: "What number, when multiplied by itself, equals this number under the square root?" For example, the square root of 144 is 12 because \( 12 \times 12 = 144 \). The square root function is essential in mathematics, used in everything from basic arithmetic to complex calculus.

To simplify a square root, you often look for perfect squares within the number. If a number isn't a perfect square, you might end up with an irrational number (like \( \sqrt{2} \)), which cannot be expressed as a simple fraction. However, for perfect squares like 144, simplifying the square root is a straightforward task.

• Look for a perfect square inside the radical.
• Replace the square root of that perfect square with its whole number value.

In conclusion, mastering square roots and recognizing which numbers are perfect squares simplifies dealing with radical expressions in math.