Problem 46 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\frac{\sqrt{2 x^{3}}}{\sqrt{9 y}}\)

Step-by-Step Solution

Answer

\(\frac{x\sqrt{2x}}{3\sqrt{y}}\)

Step 1: Simplify the Numerator and Denominator

Start by simplifying the square root expressions in both the numerator and denominator. The numerator is \(\sqrt{2x^3}\), which can be broken into \(\sqrt{2} \times \sqrt{x^3}\). The denominator is \(\sqrt{9y}\), which can be simplified to \(\sqrt{9} \times \sqrt{y} = 3\sqrt{y}\).

Step 2: Simplify \(\sqrt{x^3}\)

Simplify \(\sqrt{x^3}\) by breaking it down into \(\sqrt{x^2 \cdot x}\), which is \(\sqrt{x^2} \times \sqrt{x}\). Since \(\sqrt{x^2} = x\), it simplifies to \(x\sqrt{x}\). So, \(\sqrt{2x^3} = \sqrt{2} \times x \times \sqrt{x} = x\sqrt{2x}\).

Step 3: Combine and Simplify the Entire Expression

Now combine the simplified versions of the numerator and the denominator: \(\frac{x\sqrt{2x}}{3\sqrt{y}}\). This is the simplified radical form, as no further simplification of the square roots is possible.

Key Concepts

Square Root Simplification

Square root simplification involves breaking down the square roots into their simplest form. To do this, we identify perfect squares within the number under the square root.
This process helps in reducing the expression into something more manageable.

• The expression \(\sqrt{2x^3}\) can be simplified by breaking it down into smaller parts.
• Identify \(x^3\) as \(x^2 \cdot x\), then separate into \(\sqrt{x^2} \times \sqrt{x}\).
• The number 2 does not have a perfect square factor other than 1, so when simplifying \(\sqrt{2}\), it remains \(\sqrt{2}\).
• Finally, combine these simplifications to obtain the simplest radical form.

Numerator and Denominator Simplification

Simplifying the numerator and denominator separately is a crucial step. It ensures the overall expression is in its simplest form before combining them.
This allows us to work with simpler components.

• For the numerator \(\sqrt{2x^3}\), simplify first as \(\sqrt{2} \times x \times \sqrt{x}\), given that \(\sqrt{x^2} = x\).
• For the denominator, the expression \(\sqrt{9y}\) breaks down into \(\sqrt{9} \times \sqrt{y} = 3\sqrt{y}\) since \(\sqrt{9} = 3\).

These simplifications set the stage for further reducing the whole expression.
The goal is to simplify as much as possible while maintaining the equality of the original expression.

Expressions with Radicals

Expressions with radicals often appear complex, but analyzing them piece by piece makes them easier to handle. Radicals signify the root of a number, and simpler radicals contribute to clearer, more concise expressions.Here's a breakdown:

• Expressions like \(\frac{\sqrt{2x^3}}{\sqrt{9y}}\) involve rewriting radicals in simpler forms before performing any arithmetic operations.
• Once each radical is simplified individually, combining them follows naturally.

Keeping this in mind helps you view radicals not as hurdles in your calculations, but as elements that can often be neatly reduced to enhance clarity.

Positive Real Numbers

Whenever dealing with radicals, it is important to assume that the variables represent positive real numbers. This assumption is crucial because it allows square roots to be well-defined and ensures we only deal with real number solutions.

• Positive real numbers simplify the calculation of expressions with radicals as every root legally exists in the real number system.
• Ensuring all variables are positive avoids complications from negative roots, which may introduce imaginary numbers.

By having this foundational knowledge, your work will be more straightforward.
It also reinforces the correctness of simplified expressions and sustains consistency across various mathematical computations.