Problem 29 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{\frac{9}{50 x y^{7}}} $$

Step-by-Step Solution

Answer

The simplest form is \(\frac{3\sqrt{y}}{5y^4\sqrt{2x}}\).

Step 1: Rationalize the denominator

We need to ensure that the denominator is a rational number. To do this, multiply the numerator and the denominator by the appropriate radical to eliminate the radicals in the denominator. In this case, we multiply by \(\sqrt{y}\) since \(y^7\) can be simplified as \(y^6 \cdot y\). This gives us: \[ \frac{\sqrt{9} \cdot \sqrt{y}}{\sqrt{50xy^7} \cdot \sqrt{y}} = \frac{\sqrt{9y}}{\sqrt{50xy^8}} \]

Step 2: Simplify the radicals

Now, simplify both the numerator and the denominator by breaking them into perfect squares. The numerator becomes \(3\sqrt{y}\) since \(\sqrt{9y} = 3\sqrt{y}\). The denominator becomes \(\sqrt{50} \cdot x^{1/2} \cdot y^4\) as \(\sqrt{y^8} = y^4\). Therefore, we have: \[ \frac{3\sqrt{y}}{y^4 \cdot \sqrt{50x}} \]

Step 3: Simplify the square roots further

Simplify the \(\sqrt{50x}\) term. Since \(\sqrt{50} = 5\sqrt{2}\), we have: \[ \frac{3\sqrt{y}}{y^4 \cdot 5\sqrt{2x}} = \frac{3\sqrt{y}}{5y^4 \sqrt{2x}} \]

Step 4: Finalize the simplest form

Combine all simplified terms: the expression in simplest radical form is \[ \frac{3\sqrt{y}}{5y^4\sqrt{2x}} \]. This expression represents the simplest form of the given radical.

Key Concepts

Rationalize the Denominator

Rationalizing the denominator is a key concept when dealing with fractions that have radicals in the denominator. This process involves adjusting the fraction so that the denominator is a rational number, meaning it doesn't contain any square roots or irrational numbers. In particular, if you encounter a radical like \( \sqrt{b} \) in the denominator, you multiply both the numerator and the denominator by \( \sqrt{b} \). This step eliminates the radical from the denominator because \( \sqrt{b} \cdot \sqrt{b} = b \).

In our exercise, we started with \( \sqrt{\frac{9}{50xy^7}} \). To rationalize this expression, we noticed that \( y^7 = y^6 \cdot y \). By multiplying both the numerator and denominator by \( \sqrt{y} \), we effectively eliminated much of the radical from the denominator. This step is crucial as it simplifies the entire fraction, making it easier to work with.

Square Roots

Understanding square roots is essential for simplifying radical expressions. A square root of a number \( a \) is a value \( b \) such that \( b^2 = a \). Simply put, \( \sqrt{a} \) is a number that, multiplied by itself, gives \( a \).

In the exercise, we dealt with a square root in the expression \( \sqrt{9} \). Since \( 9 \) is a perfect square, it simplifies to \( 3 \) because \( 3^2 = 9 \). Knowing how to simplify square roots, especially those with perfect squares, helps reduce complex radicals to simpler forms. This fundamental concept is very handy for further steps, like simplifying the numerator or denominator.

Simplifying Radicals

Simplifying radicals involves breaking down radical expressions into simpler terms by identifying and extracting perfect squares. This process can make calculations more straightforward and help transform complex expressions.

For instance, in our simplified expression, the square root in the numerator, \( \sqrt{9y} \), was simplified to \( 3\sqrt{y} \). Similarly, in the denominator, \( \sqrt{y^8} \) simplified to \( y^4 \), since \( (y^4)^2 = y^8 \). Moreover, \( \sqrt{50} \) was reduced to \( 5\sqrt{2} \). By consistently looking for factors of the radicand that are perfect squares, you streamline the expression and bring it to its simplest radical form.

Fractional Radicals

Fractional radicals are expressions where radicals are part of a fraction, usually as the numerator or the denominator. Handling these requires a blend of strategies, such as rationalization and radical simplification, to manage both parts of the fraction effectively.

In addressing the given fraction \( \frac{\sqrt{9y}}{5y^4\sqrt{2x}} \), each component was carefully simplified. The process involved rationalizing the denominator initially, and then simplifying each radical expression. It required a thorough understanding of how to manipulate radicals both within the fraction and as standalone components.

• This approach ensures the overall expression remains accurate while promoting clarity.
• A common goal is to remove radicals from the denominator, as is standard in math for presenting expressions clearly.

By mastering fractional radicals, you not only solve existing problems but also gain confidence in tackling more complex equations.