Problem 22 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{48}\)

Step-by-Step Solution

Answer

\( \sqrt{48} = 4\sqrt{3} \)

Step 1: Identify a Perfect Square Factor

The first step is to find a perfect square that can divide the number 48. In this case, 16 is a perfect square that divides 48 evenly since \( 16 \times 3 = 48 \).

Step 2: Rewrite the Radical Using Perfect Squares

Express the number under the square root, 48, as a product of 16 and 3: \( \sqrt{48} = \sqrt{16 \times 3} \).

Step 3: Apply the Property of Square Roots

Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to simplify the expression: \( \sqrt{48} = \sqrt{16} \times \sqrt{3} \).

Step 4: Simplify the Square Root Expression

Calculate the square root of the perfect square, \( 16 \), to simplify the expression: \( \sqrt{16} = 4 \). So the expression becomes \( 4\sqrt{3} \).

Key Concepts

Square Roots

A square root is essentially a value that, when multiplied by itself, gives you the original number. Think of it as the opposite of squaring a number. For example, if you square the number 5, you get 25, which means the square root of 25 is 5.

When dealing with square roots, especially in algebraic settings, it is useful to simplify them as much as possible for ease of calculation and interpretation. Simplifying involves breaking down a non-perfect square into any perfect square factors it might have, which makes the arithmetic easier to handle.

• Perfect squares such as 1, 4, 9, 16, etc., offer a simpler approach to calculation.
• The square root of these numbers, say \(\sqrt{16}\) which equals 4, is whole and straightforward to handle.

By understanding how to break down larger numbers using their square root properties, one can simplify complex expressions into more manageable parts.

Perfect Squares

Perfect squares are numbers that are the square of whole numbers. They hold particular significance in simplifying radicals because they provide a neat, whole-value result when their square root is taken.

For instance:

• 4 is a perfect square, since \(2 \times 2 = 4\).
• 9 is also a perfect square, arising from \(3 \times 3 = 9\).
• 16 gives 4 when square rooted, as discussed in the original exercise when simplifying \(\sqrt{48}\).

In algebra, identifying perfect squares can help in various tasks including factoring and simplifying expressions. Knowledge of perfect squares is handy for quickly recognizing smaller factors applicable in larger numbers, which simplifies the calculations drastically by removing decimals and fractions from square root problems.

Algebraic Expressions

Algebraic expressions involve variables, constants, and coefficients. When working with radicals, they often appear in equations where simplification is required to either solve or further manipulate the expression. Simplifying radicals within algebraic expressions can aid in the ease of solving equations.

Let's consider simplifying algebraic expressions involving square roots. Suppose you have an expression like \(\sqrt{a^2}\), knowing that it simplifies directly to \(a\) if \(a\) is non-negative. This kind of simplification helps in solving more complex algebraic problems.

• Simplifying expressions makes equations easier to solve, allowing for more straightforward solutions.
• Radicals are often integrated with variables, thus understanding square roots and perfect squares is beneficial.

Using the properties of square roots and the recognition of perfect squares in longer expressions breaks complicated relations into manageable parts, thereby facilitating algebraic manipulation. Understanding the basic properties of these elements is crucial for efficiently working with equations in algebra.