Problem 17 - Algebra 2 and Trigonometry

Question

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \frac{\sqrt{18 c^{3}}}{\sqrt{9 c}} $$

Step-by-Step Solution

Answer

The simplest form is \( c\sqrt{2} \).

Step 1: Simplify the Radicals

First, break down each square root into simpler components. Start with the numerator: \( \sqrt{18c^3} \). This can be rewritten by factoring: \( \sqrt{18c^3} = \sqrt{18} \times \sqrt{c^3} \). Similarly, break the denominator as \( \sqrt{9c} = \sqrt{9} \times \sqrt{c} \).

Step 2: Evaluate Inside the Radicals

Simplify the purely numerical parts and split off perfect squares. For \( \sqrt{18} \), note that \( 18 = 9 \times 2 \) so \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \). Likewise, for \( \sqrt{9} = 3 \). The \( \sqrt{c^3} \) can be broken into \( c \times \sqrt{c} \) because \( c^3 = c^2 \times c \) and \( \sqrt{c^2} = c \).

Step 3: Substitute Back into the Fraction

Using the simplified forms from Step 2, substitute back: \( \frac{\sqrt{18c^3}}{\sqrt{9c}} = \frac{3\sqrt{2} \times c \sqrt{c}}{3\times \sqrt{c}} \).

Step 4: Simplify the Fraction

Cancel out common factors from the numerator and denominator. Cancel \( 3 \) and \( \sqrt{c} \) from the numerator and denominator: \( \frac{3c\sqrt{2c}}{3\sqrt{c}} = c\sqrt{2} \).

Step 5: Finalize the Answer

Conclude the simplification by acknowledging that the fraction has been completely simplified. The final answer is \( c\sqrt{2} \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, often represented by the radical symbol \( \sqrt{} \). When you see this symbol, it essentially means, "What number squared gives me this number?" Understanding square roots can be crucial, especially when simplifying more complex mathematical expressions. For instance, in the problem involving \( \sqrt{18} \), we want to find a number that, when squared, equals 18. However, 18 is not a perfect square. Instead, we break it down into factors that are perfect squares, such as 9, because \( 9 \times 2 = 18 \). This gives us:

• \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \)

This method helps in simplifying expressions and solving equations involving radicals.

Fraction Simplification

Fraction simplification makes expressions easier to work with by reducing them to their simplest form. In the given exercise, we have a fraction \( \frac{\sqrt{18c^3}}{\sqrt{9c}} \). To simplify, you need to recognize and cancel out common factors within a fraction.

The approach is to first simplify each part of the fraction separately. The numerator and denominator are broken down into simpler radical terms like:

• Numerator: \( \sqrt{18c^3} = 3c\sqrt{2c} \)
• Denominator: \( \sqrt{9c} = 3\sqrt{c} \)

After breaking down the components, any common factors, like the 3 and \( \sqrt{c} \), can be cancelled out. This leaves the expression in its simplest form, which is streamlined and easier to handle mathematically. The process makes complex fractions more manageable, aiding in both practical computation and theoretical analysis.

Radical Expressions

Working with radical expressions involves simplifying terms that contain square roots or other roots. These types of expressions are often seen in mathematical problems, requiring a good grasp of simplification techniques. To manipulate radical expressions effectively, consider the following steps:

• Identify and split components of the radicals specifically into perfect squares or cubes, such as \( c^3 = c^2 \times c \).
• Apply rules of radicals like \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \) and separate them into more manageable terms.
• Combine like terms when possible, eliminating radicals by taking terms out of the square root when they are indeed a perfect square, for example, \( \sqrt{c^2} = c \).

By simplifying these radical expressions, as seen in the exercise with \( \frac{\sqrt{18c^3}}{\sqrt{9c}} \), it becomes easier to solve and further explore mathematical concepts. It's an essential skill in mathematics, applicable in more advanced topics, making complex calculations accessible and more straightforward.