Problem 18 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{192} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 8\sqrt{3} \).

Step 1: Identify Perfect Squares

First, identify any perfect square factors of 192. A perfect square is a number whose square root is an integer. In this case, 192 can be broken down into the product of smaller numbers. Notice that 192 = 64 × 3, where 64 is a perfect square.

Step 2: Simplify the Square Root of the Perfect Square

Take the square root of the perfect square factor 64. Since 64 is a perfect square, its square root is an integer: \( \sqrt{64} = 8 \).

Step 3: Express the Original Square Root in Simplified Form

Substitute the square root of the perfect square back into the original expression. Thus, \( \sqrt{192} \) becomes \( \sqrt{64 \times 3} = \sqrt{64} \times \sqrt{3} = 8\sqrt{3} \).

Step 4: Final Simplified Expression

The simplified form of \( \sqrt{192} \) is \( 8\sqrt{3} \). Ensure that no further simplification is possible, confirming that \( 3 \) is not a perfect square, and thus \( \sqrt{3} \) cannot be simplified further.

Key Concepts

Perfect Squares

When dealing with square roots, understanding **perfect squares** is essential. A perfect square is a number that results from squaring an integer. In simpler terms, if we multiply an integer by itself, we get a perfect square. For instance, 1, 4, 9, 16, 25, and 64 are perfect squares because:

• \( 1 = 1 \times 1 \)
• \( 4 = 2 \times 2 \)
• \( 9 = 3 \times 3 \)
• \( 16 = 4 \times 4 \)
• \( 25 = 5 \times 5 \)
• \( 64 = 8 \times 8 \)

Recognizing perfect squares is the first thing to do when you aim to simplify a square root. In our exercise, we have the number 192. By breaking it down, we found that 192 can be expressed as 64 times 3. Here, 64 is the perfect square, simplifying our simplification process greatly.

Square Root Properties

To simplify expressions involving square roots, it's helpful to know some basic **square root properties**. The square root operation is based on inverse squaring. There are some key rules you should remember:

• \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)
• \( \sqrt{a^2} = a \) when \( a \geq 0 \)

These properties allow us to split a square root into more manageable parts.
In our example, \( \sqrt{192} \) becomes \( \sqrt{64 \times 3} \), which can be rewritten using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Therefore, it transforms into \( \sqrt{64} \times \sqrt{3} \).
Using the fact that \( 64 = 8^2 \), we simplify \( \sqrt{64} \) to 8. This transformation relies heavily on these handy properties, making simplification possible.

Simplified Expression

A **simplified expression** ensures that a square root has been reduced to its most basic form. After recognizing perfect squares and employing square root properties, you arrive at the simplest version of the expression.
Let's look at our example: after breaking down \( \sqrt{192} \) into \( \sqrt{64} \times \sqrt{3} \) and calculating \( \sqrt{64} = 8 \), we end up with \( 8\sqrt{3} \).
This final expression, \( 8\sqrt{3} \), cannot be simplified further because 3 is not a perfect square, and thus, \( \sqrt{3} \) remains in the expression. Simplified expressions are valuable in mathematics because they make numbers easier to work with and understand, revealing the cleanest form of an expression.