Problem 31 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{144} $$

Step-by-Step Solution

Answer

The square root of 144 is 12.

Step 1: Understand the Problem

First, it is crucial to understand what simplifying a square root means. The square root of a number x is the number that, when multiplied by itself, gives x as the result. So, the task is to find that number for 144.

Step 2: Calculate the Square Root

The next step is to calculate the square root of 144. We need to find a number that, when multiplied by itself, equals 144. The number that meets this requirement is 12 because \(12*12 = 144\).

Step 3: Write Down the Answer

After we have found the number that, when squared, gives us 144, we can write it down as the answer.

Key Concepts

Square Root Calculation

Calculating the square root of a number is a foundational skill in mathematics that allows you to simplify expressions involving square roots. A square root, symbolized by the radical sign \( \sqrt{} \), undoes the operation of squaring a number. For instance, if you square the number 5, you get 25. The square root of 25 is thus 5, because \(5 \times 5 = 25\).

In practice, the square root calculation often involves finding the largest perfect square factor of the number under the radical sign and simplifying from there. To illustrate, let's consider a concrete example:\[ \sqrt{144} \] Here, 144 is a perfect square (because 12 squared is 144), so the square root calculation simplifies to 12. In general, when the number under the radical is not a perfect square, you can use the prime factorization method or an estimation method to simplify the square root as much as possible. For larger numbers, calculators are a convenient tool for finding square roots quickly.

Radical Expressions

Radical expressions involve numbers under the root symbol. In the realm of radicals, expressions can have square roots, cube roots, or higher-order roots. Your main goal with radical expressions is to simplify them, which includes reducing any square roots to their simplest form.

When simplifying radical expressions, it's important to:

• Identify perfect squares within the radical,
• Break down the expression using the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), and
• Reduce any fractions inside the radical to their lowest terms.

Always look for opportunities to simplify a radical expression by removing perfect squares, which makes the expression cleaner and easier to use in subsequent calculations. If the number under the square root is a perfect square, the radical simplifies entirely, without a radical sign in the final answer, as was the case with \(\sqrt{144}\).

Perfect Squares

Understanding perfect squares is crucial for simplifying square roots. A perfect square is a number that's the product of an integer multiplied by itself. For example, 9 is a perfect square since \(3 \times 3 = 9\).

Here are some properties of perfect squares:

• They are always non-negative.
• Every perfect square has an odd number of positive factors.
• A perfect square ends in 0, 1, 4, 5, 6, or 9 in the decimal system.

Recognizing perfect squares helps you simplify square roots quickly because the square root of a perfect square is always an integer. With numbers that are not perfect squares, you might get an irrational number, which cannot be expressed as a simple fraction, and you'll need to approximate the value instead. Knowing the common perfect squares (such as 1, 4, 9, 16, 25, ...) aids in identifying when a square root can be easily evaluated, as in the example \(\sqrt{144} = 12\), since 144 is a perfect square.