Problem 29 - College Algebra

Question

Use the quotient rule to simplify the expressions in Exercises. Assume that \(x>0.\) $$\frac{\sqrt{150 x^{4}}}{\sqrt{3 x}}$$

Step-by-Step Solution

Answer

The simplified version of the given expression \(\frac{\sqrt{150 x^{4}}}{\sqrt{3 x}}\) using the quotient rule is \(5x\sqrt{2}\).

Step 1: Rewrite the expression using the quotient rule

Apply the quotient rule for radicals on the given expression which says that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). The expression becomes: \(\frac{\sqrt{150 x^{4}}}{\sqrt{3 x}}.\)

Step 2: Simplify the square roots separately

Here, separate the factors under the radical that have perfect square roots and simplify: \(\sqrt{150 x^{4}} = \sqrt{25*6*\left(x^{2}\right)^{2}} = 5x^{2}\sqrt{6}\) and \(\sqrt{3 x} = \sqrt{3*x^{1}} = x\sqrt{3}\). So, the expression becomes \(\frac{5x^{2}\sqrt{6}}{x\sqrt{3}}\).

Step 3: Reduce the expression to lowest term

In this step, the goal is to cancel out common terms in the numerator and the denominator. On simplifying, \(\frac{5x^{2}\sqrt{6}}{x\sqrt{3}} = 5x\sqrt{\frac{6}{3}} = 5x\sqrt{2}\).

Key Concepts

Radical Expressions

Radical expressions involve roots, most commonly square roots. When working with radicals, the key is to understand how to manipulate them in the form of an expression. For example, in the problem, we deal with

• The square root of a number: \(\sqrt{150x^4}\).
• The radical part of a fraction: \(\frac{\sqrt{a}}{\sqrt{b}}\).

A valuable technique with radicals is the quotient rule for square roots, which allows us to simplify the division of two square roots into a single radical. This process is critical when you aim to simplify expressions by finding and canceling common terms in the numerator and the denominator. The quotient rule helps to reframe the function in a single square root: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). This makes it manageable to further simplify.
Radical expressions can look complicated, but understanding the role of factors under the radicals allows us to find perfect squares and ease the simplification process.

Simplification

Simplification is the process of reducing an expression to its simplest form. In math, this often means eliminating radicals or reducing them to easy-to-manage terms. When simplifying, start by splitting the factors inside the radicals into perfect squares. For example:

• The number \(150\) can be split into \((25 \times 6)\), so \(\sqrt{150} = 5\sqrt{6}\).
• Similarly, \(x^4\) can be broken down to \(\left(x^2\right)^2\), simplifying to \(x^2\).

We then recombine these simplified components. In our problem, we turn \(\sqrt{150x^4}\) into \(5x^2\sqrt{6}\). Simplification is about recognizing these opportunities to break down numbers and variables to make them manageable.
The last step in simplification involves reducing the rational expression by canceling common terms in the numerator and the denominator, such as reducing \(\frac{5x^2}{x} = 5x\). This results in a cleaner and more concise version of the original expression.

Square Roots

Square roots ask the question, "What number multiplied by itself gives this value?" For example, the square root of \(25\) is \(5\) because \(5 \times 5 = 25\).
In expressions with variables, like \((x^4)\), the square root works similarly but focuses on powers. The square root of \(x^4\) is \(x^2\) because \((x^2 \times x^2 = x^4)\). When dealing with non-perfect squares like \(\sqrt{150}\), we look for the largest perfect square factor, which is \(\sqrt{25}\) in our case.
Breaking down and understanding the underlying factors allows simplification of square roots even if the term isn't perfect.

• Identify parts that are perfect squares. For \(150\), this is \(25\).
• Simplify. \(\sqrt{25} = 5\).

Utilizing such strategies simplifies handling square roots both numerically and with variables. This specific task is often fundamental in calculus and algebra while dealing with complex expressions.