Problem 59 - Algebra for College Students

Question

Change each radical to simplest radical form. \(-\frac{4 \sqrt{12}}{\sqrt{5}}\)

Step-by-Step Solution

Answer

The simplest form is \( -\frac{8\sqrt{15}}{5} \).

Step 1: Express Inside Radical Factors

The first step is to simplify the radical expression inside the numerator. The expression in the numerator is \( \sqrt{12} \). We recognize that 12 can be factored into 4 and 3, where 4 is a perfect square. So, \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \). Now the numerator becomes \( -4 \times 2\sqrt{3} = -8\sqrt{3} \).

Step 2: Rationalize the Denominator

The next step is to rationalize the fraction by eliminating the square root from the denominator. We have the expression \( -\frac{8\sqrt{3}}{\sqrt{5}} \). To rationalize it, multiply both the numerator and denominator by \( \sqrt{5} \) to get: \[ -\frac{8\sqrt{3} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = -\frac{8\sqrt{15}}{5} \].

Step 3: Simplify the Expression

Now, verify if the expression can be simplified further. As \( -\frac{8\sqrt{15}}{5} \) is already in its simplest form and cannot be further reduced, we can conclude the process here.

Key Concepts

Simplifying Radicals

When dealing with radicals, especially in mathematics involving square roots, simplifying them is a key skill to make the expressions more manageable. The process of simplifying radicals involves expressing a radical so that the expression inside has no perfect square factors other than 1.

For example, consider the radical expression \( \sqrt{12} \). The number 12 is not a perfect square, but it can be broken down into factors. Specifically, 12 can be factored as \( 4 \times 3 \), where 4 is a perfect square. This allows us to simplify \( \sqrt{12} = \sqrt{4 \times 3} \).

• Recognize \( \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} \).
• Since \( \sqrt{4} = 2 \), we have \( 2 \sqrt{3} \).

This transformation is what we call simplifying the radical. By breaking down the number into its simplest components, we're left with expressions like \( 2\sqrt{3} \) instead of \( \sqrt{12} \). This method not only simplifies calculations but is often necessary for further operations like adding or multiplying radicals.

Perfect Squares

Understanding perfect squares is essential in simplifying radicals and rationalizing denominators. A perfect square is a number that is the square of an integer. For example, numbers like 1, 4, 9, 16, 25, and so on are perfect squares.

Perfect squares come in handy when simplifying expressions under a square root. If you can identify a part of a number as a perfect square, you can simplify the expression. In our example with \( \sqrt{12} \), we identified 4 as a perfect square. This identification is key because it allows us to simplify the radical by extracting the square root of this perfect square:

• Since \( 4 \) is \( 2^2 \), then \( \sqrt{4} = 2 \).
• Replace \( \sqrt{4} \) in the expression: \( \sqrt{12} = 2\sqrt{3} \).

Using perfect squares helps simplify complex expressions, making them less cumbersome for further algebraic manipulations, such as in rationalizing denominators.

Simplifying Fractions

Simplifying fractions often involves reducing the numerator and the denominator to smaller numbers while maintaining the same overall value of the fraction. In expressions involving radicals, this can also mean removing the square root from the denominator, a process known as "rationalizing the denominator."

In the expression \( -\frac{8\sqrt{3}}{\sqrt{5}} \), the square root in the denominator can be removed by multiplying both the numerator and denominator by the same radical found in the denominator, making use of the property \( \sqrt{a} \times \sqrt{a} = a \).

• Multiply by \( \sqrt{5} \): \( -\frac{8\sqrt{3} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} \).
• This results in \( -\frac{8\sqrt{15}}{5} \), a fraction with a rationalized denominator.

The fraction \( -\frac{8\sqrt{15}}{5} \) is now simplified and easier to handle. Simplified fractions are critical for solving equations and further calculations, ensuring expressions are kept as simple as possible while retaining equivalence.