Problem 7 - 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. $$ \sqrt{24 x^{2}} \div \sqrt{3 x^{3}} $$

Step-by-Step Solution

Answer

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

Step 1: Apply Quotient Rule for Radicals

Use the quotient rule for radicals, which states that \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Apply it to the problem: \( \frac{\sqrt{24x^2}}{\sqrt{3x^3}} = \sqrt{\frac{24x^2}{3x^3}} \).

Step 2: Simplify the Fraction Inside the Radical

Simplify the expression inside the square root: \( \frac{24x^2}{3x^3} = \frac{24}{3} \times \frac{x^2}{x^3} = 8 \times \frac{1}{x} = \frac{8}{x} \).

Step 3: Simplify the Radical Expression

Now simplify \( \sqrt{\frac{8}{x}} \). Recognize that \( 8 = 2^3 \), so \( \sqrt{8} = \sqrt{2^3} = 2\sqrt{2} \). Thus, \( \sqrt{\frac{8}{x}} = \frac{2\sqrt{2}}{\sqrt{x}} \).

Step 4: Rationalize the Denominator

To eliminate the square root in the denominator, multiply numerator and denominator by \( \sqrt{x} \). This gives \( \frac{2\sqrt{2}}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{2\sqrt{2x}}{x} \).

Key Concepts

Quotient Rule for Radicals

When faced with a problem that involves dividing two radicals with the same index, the quotient rule for radicals becomes very handy. This rule tells us that dividing two square roots is equivalent to taking the square root of the division of their radicands. Here's how it works:

• If you have two expressions under a square root, such as \( \frac{\sqrt{a}}{\sqrt{b}} \), you can combine them under a single square root \( \sqrt{\frac{a}{b}} \).
• This eliminates the need to manually divide the numbers outside of the radical, simplifying your work.

For example, if you start with \( \sqrt{24x^2} \div \sqrt{3x^3} \), the quotient rule lets you rewrite this as \( \sqrt{\frac{24x^2}{3x^3}} \). This formula is not only simple but also crucial in easing the process.

Rationalizing the Denominator

Rationalizing the denominator means getting rid of any radicals in the denominator of a fraction. This is often required because expressions are considered simpler when the denominator is rational. Let's break this down:

• If you have a fraction with a square root in the denominator like \( \frac{1}{\sqrt{x}} \), you multiply the numerator and the denominator by the same square root. This gives you \( \frac{\sqrt{x}}{\sqrt{x} \times \sqrt{x}} = \frac{\sqrt{x}}{x} \).
• The purpose is to "cancel out" the square in the denominator, effectively converting it into a whole number.

In the original exercise, after simplifying the inside of the radical to \( \sqrt{\frac{8}{x}} \), rationalizing it by multiplying by \( \sqrt{x} \) gives us \( \frac{2\sqrt{2x}}{x} \). It's a key step to move towards a fully simplified expression that’s easier to understand and use.

Simplifying Expressions with Variables

Simplifying expressions with variables involves reducing the expression to its most basic form. This process often includes canceling like terms, using exponent rules, and factoring. Here's how you can simplify expressions with variables:

• First, examine coefficients and variables separately. For the coefficient itself, divide the numbers as usual. In our example, \( \frac{24}{3} \) simplifies to \( 8 \).
• For the variables, apply the law of exponents: \( \frac{x^m}{x^n} = x^{m-n} \). In the example, \( \frac{x^2}{x^3} = x^{2-3} = x^{-1} = \frac{1}{x} \).
• Once the fraction inside the radical is simplified, check to see if further simplification of the radical is possible by looking at perfect squares or factoring base numbers.

The example solution simplifies to \( \sqrt{\frac{8}{x}} \) leading to \( \frac{2\sqrt{2x}}{x} \) where the expression is reduced to its simplest form, keeping it easy to handle in mathematical operations.