Problem 15 - 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. $$ \frac{7}{\sqrt{7 y}} $$

Step-by-Step Solution

Answer

The simplest form is \(\frac{\sqrt{7y}}{y}\).

Step 1: Rationalize the Denominator

To simplify the fraction \(\frac{7}{\sqrt{7y}}\), we need to eliminate the square root from the denominator. This is done by multiplying both the numerator and denominator by \(\sqrt{7y}\). The equation becomes: \[ \frac{7 \cdot \sqrt{7y}}{\sqrt{7y} \cdot \sqrt{7y}} = \frac{7\sqrt{7y}}{7y} \]

Step 2: Simplify the Expression

In the new expression \(\frac{7\sqrt{7y}}{7y}\), the 7 in the numerator and denominator cancels out, simplifying to: \[ \frac{\sqrt{7y}}{y} \]

Step 3: Simplify Further if Possible

Since no further simplification is possible, and variables in the radicand are non-negative, the simplest form of the original expression is maintained. Thus, the simplest form of the fraction is \(\frac{\sqrt{7y}}{y}\).

Key Concepts

Simplifying Radical Expressions

When working with radical expressions involving fractions, it's crucial to first pinpoint the radicals that may complicate the expression. The goal is to eliminate these radicals, especially from the denominator, which paves the way to simplification.
Radical expressions often include square roots, cube roots, or any other root that must be managed. A core tactic is to rationalize the denominator, meaning that you aim to clear the radical by multiplying the expression by a strategic value—in this scenario, it's by a similar root present in the denominator.

• Start by identifying radical expressions that you need to address, focusing mainly on those in the denominator.
• Use a multiplication technique to remove the root from the denominator by introducing a similar radical into the numerator and denominator.
• Simplify the resulting expression by handling coefficients and variables correctly.

Simplifying these radicals correctly will make the expression more manageable, and easier to interpret in further equations.

Square Roots

Square roots represent numbers that, when multiplied by themselves, return the original number under the root sign. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9.
In fractional expressions, square roots commonly appear as we work to simplify expressions into their most basic rational forms. It's important to handle square roots correctly to maintain the integrity of the expression.

• Recognize square roots as they appear either as whole numbers or with variables within radical signs.
• Square roots located in the denominator need to be eliminated. This is achieved through the process of rationalization, which involves multiplying by an equivalent square root.
• Understand that multiplying the square root by itself cancels the root, since \( \sqrt{x} \times \sqrt{x} = x \).

Mastering square roots ensures you can tackle more complex expressions without stumbling on the basics. The rule of thumb: simplify step by step.

Simplifying Fractions

Simplifying fractions is about breaking down an expression to its most straightforward form. After eliminating radicals from the denominator, simplifying involves reducing the fraction as much as possible.
Whether you're dealing with numbers alone or variables, keeping fractions simple leads to easier calculations and better comprehension of mathematical expressions.

• Begin by ensuring that there are no radicals in the denominator.
• Look for common factors in the numerator and denominator. These could be numbers or variables with similar exponents.
• Cancel out these common factors to reduce the fraction to its lowest terms.

In our example, eliminating the number 7 from both the numerator and denominator simplified the expression to its simplest form: \( \frac{\sqrt{7y}}{y} \). Keeping expressions simple is a cornerstone of clear mathematical problem-solving.