Problem 30 - College Algebra

Question

Use the quotient rule to simplify the expressions in Exercises. Assume that \(x>0.\) $$\frac{\sqrt{24 x^{4}}}{\sqrt{3 x}}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \( 2x\sqrt{2x} \)

Step 1: Simplify the numerator

In the numerator, we have \(\sqrt{24 \cdot x^{4}}\). As stated in the analysis, we can treat the square root of a product of numbers as the product of the square roots of the individual numbers. Hence, \(\sqrt{24 \cdot x^{4}} = \sqrt{24} \cdot \sqrt{x^{4}}\). We simplify \(\sqrt{24}\) to get 2\(\sqrt{6}\) and simplify \(\sqrt{x^{4}}\) to get \(x^{2}\). Hence, \(\sqrt{24 \cdot x^{4}} = 2x^{2}\sqrt{6}\).

Step 2: Simplify the denominator

Next is the denominator, \(\sqrt{3 \cdot x}\). Similarly, \(\sqrt{3 \cdot x}\) equals to \(\sqrt{3}\cdot\sqrt{x}\). Hence, the denominator remains as \(\sqrt{3x}\). The given expression then becomes \( \frac {2x^{2}\sqrt{6}} {\sqrt{3x}} \)

Step 3: Cancel out like terms

Using the rule that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), the given expression can be simplified to \( \frac{2x^{2}\sqrt{6}}{\sqrt{3}\sqrt{x}} = 2x^{2} \sqrt{\frac{6}{3x}} \). Simplify further to get \( 2x\sqrt{2x} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making an expression easier to work with. You deal with terms and try to reduce them to their most basic form. In mathematics, simplicity is king because it makes calculations easier and reduces errors.
Let's break down the process:

• Identify like terms: These are terms that have the same variable raised to the same power. For example, in the expression \(2x^2 + 4x^2\), you can combine these to get \(6x^2\).
• Factor common terms: Look for common factors in an expression, like in \(6x + 3 = 3(2x + 1)\).
• Use arithmetic to simplify: Sometimes, it's as simple as doing the arithmetic, as in \(4 + 2 = 6\).

The benefit of simplifying expressions is that it leads to more manageable numbers and terms. Once simplified, complex problems become less intimidating.

Radical Expressions

Radical expressions involve roots, like square roots or cube roots. These can be tricky at first, but with practice, they become easier to handle. Understanding their properties is vital to simplifying them.

• Radicals can be rewritten for easier manipulation. For instance, \(\sqrt{a \cdot b} = \sqrt{a}\sqrt{b}\). This property was crucial in the original problem where the square root of a product was simplified by separate square roots.
• Simplifying the radicand: If the radicand (the number inside the root) can be factored into a perfect square, the radical can be simplified. For example, \(\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}\).
• Reducing radicals in fractions: A quotient involving radicals can also be simplified, \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), making it easier to work with fractional radicals.

Understanding these properties helps in manipulating and simplifying radical expressions so they become easier to handle, just like in the exercise when simplifying \(\sqrt{24x^4}\) into \(2x^2\sqrt{6}\).

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Simplifying these fractions often involves several steps to ensure you are working with the simplest form.

• Factor both numerator and denominator: This is key to simplifying an algebraic fraction. If \(\frac{2x^2 + 4x}{x} \), you factor out \(x\) from the numerator to simplify the expression.
• Cancel out common terms: Once factored, common factors in the numerator and denominator can be cancelled. This reduces the fraction to its simplest form.
• Handle complex fractions: If a fraction is complex, meaning it has a fraction in its numerator, denominator, or both, simplify step by step. Simplify the numerator and denominator separately before dividing.

Understanding how to manipulate these algebraic expressions is vital, as these operations often appear in exams or higher-level math problems. In the context of the exercise, simplifying \(\frac{2x^2\sqrt{6}}{\sqrt{3x}}\) into \(2x\sqrt{2x}\) shows these principles in action.