Problem 37 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ \sqrt{24} $$

Step-by-Step Solution

Answer

\( \sqrt{24} = 2\sqrt{6} \).

Step 1: Express the Radicand as a Product of Perfect Squares

To simplify \( \sqrt{24} \), we first express 24 as a product of prime factors, giving us \( 24 = 2^3 \times 3 \). We aim to write 24 as a product where one of the factors is a perfect square. We can rearrange this as \( 24 = 4 \times 6 \), where 4 is a perfect square (\( 2^2 \)).

Step 2: Separate the Radicand under the Square Root

Using the property that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), we can separate the radicand such that \( \sqrt{24} = \sqrt{4 \times 6} \). This becomes \( \sqrt{4} \times \sqrt{6} \).

Step 3: Simplify the Expression

Since we know \( \sqrt{4} = 2 \), we now simplify \( \sqrt{4} \times \sqrt{6} \) to give \( 2 \times \sqrt{6} \). Therefore, \( \sqrt{24} = 2\sqrt{6} \).

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. They are crucial in simplifying radicals because they help in breaking down numbers under a square root into simpler components. A perfect square has an integer square root. For instance:

• 4 is a perfect square because \( 2 \times 2 = 4 \).
• 9 is a perfect square since \( 3 \times 3 = 9 \).
• 16 is a perfect square because \( 4 \times 4 = 16 \).

Recognizing perfect squares makes it easier to simplify radical expressions. When simplifying the square root of a number, identifying a perfect square as a factor of the number can reduce the complexity. For example, in \( \sqrt{24} \), we identified that 4 (a perfect square) is a factor of 24, leading us to the simpler form \( 2\sqrt{6} \). This involves extracting \( \sqrt{4} = 2 \) from the square root, making the expression more approachable.

Square Roots

Square roots are operations that reverse the effect of squaring a number. The square root of a number \( x \) is a number \( y \) such that \( y^2 = x \). Simplifying square roots might involve recognizing whether the radicand contains a perfect square. In \( \sqrt{24} \), we managed to split the number into factors because square roots obey the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
This property allows separating the expression and makes it easier to handle. Let's say \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} \), simplifying further since \( \sqrt{4} \) results in 2.
This same rule applies universally, allowing the simplification of more complex expressions by breaking the numbers under the radical into manageable parts.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. This concept is fundamental to simplifying radicals, as it helps identify any hidden perfect squares. When we needed to simplify \( \sqrt{24} \), we performed a prime factorization:

• 24 can be broken down into its prime factors: \( 24 = 2 \times 2 \times 2 \times 3 \).

From this breakdown, you can group the factors to find any perfect squares. In the case of 24, \( 2 \times 2 \) (or \( 2^2 \)) appears as a perfect square.
Recognizing this allows extraction of the perfect square factor from the square root. Identifying and removing square factors simplifies the expression, leading us from the original \( \sqrt{24} \) to a more straightforward form: \( 2\sqrt{6} \). Prime factorization provides a clear view of the number's structure, making arithmetic operations simpler and more intuitive.