Problem 61 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{18 x^{4} y^{6}}{3 z}}\)

Step-by-Step Solution

Answer

Rationalizing the numerator gives \(x^{2} y^{3} \frac{\sqrt{6z}}{z}\).

Step 1: Simplify the Expression Inside the Square Root

The original expression is \(\sqrt{\frac{18 x^{4} y^{6}}{3 z}}\). First, simplify the fraction inside the square root by dividing both the numerator and the denominator by 3. This gives \(\sqrt{\frac{6 x^{4} y^{6}}{z}}\).

Step 2: Split the Square Root

Next, apply the property of square roots that allows us to split the square root of a fraction into the quotient of two square roots. This gives us \(\frac{\sqrt{6 x^{4} y^{6}}}{\sqrt{z}}\).

Step 3: Simplify the Numerator Square Root

Focus on the numerator \(\sqrt{6 x^{4} y^{6}}\). This can be further simplified by taking out the squares of the variables: \(\sqrt{6} \cdot x^{2} \cdot y^{3}\), since the square root of \(x^{4}\) is \(x^{2}\) and the square root of \(y^{6}\) is \(y^{3}\).

Step 4: Rationalize the Numerator

To rationalize the numerator \(\sqrt{6} \cdot x^{2} \cdot y^{3}\), consider multiplying both the numerator and denominator of the original expression by \(\sqrt{z}\). This transforms the expression into \(x^{2} y^{3} \frac{\sqrt{6z}}{z}\).

Step 5: Write Down the Final Result

After rationalizing the numerator, the final expression becomes \(x^{2} y^{3} \frac{\sqrt{6z}}{z}\). This is the simplified and rationalized form of the original expression.

Key Concepts

Square Roots Property

The square roots property is an essential tool in mathematics that helps simplify complex expressions and solve equations. Let's break it down into a few simple ideas:

• Separation of Roots: You can separate the square root of a fraction into two separate roots. For instance, \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). This property is quite handy when dealing with expressions in the form of a fraction under a square root.
• Breaking Down the Radicand: Another way to simplify a square root is by identifying perfect squares within the radicand (the expression inside the square root). In our original exercise, the radicand contained perfect powers of variables that could be easily extracted.
• Combining and Simplifying: After separating or breaking down expressions, you can further simplify them, which can greatly help in solving the given problem.

Simplifying Expressions

Simplifying expressions means to make them as simple as possible by performing operations like combining like terms, reducing fractions, and simplifying roots.In the exercise provided, we first simplified the fraction \(\frac{18 x^{4} y^{6}}{3 z}\) by dividing both the numerator and the denominator by 3. This resulted in \(\frac{6 x^{4} y^{6}}{z}\). Once inside the square root, simplifying continues:

• Variable Powers: Recognizing and extracting powers inside the square root, such as \(x^{4}\) becoming \(x^{2}\) and \(y^{6}\) becoming \(y^{3}\).
• Constant Simplification: Leaving as is, factors such as \( \sqrt{6} \), which can’t be simplified further as a whole number.

The goal is always to make the expression as straightforward and manageable as possible.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator contain polynomials.In the task of rationalizing the numerator, we encounter a rational expression with square roots. Here's how we work with it:

• Fraction Components: Start with the expression \(\frac{\sqrt{6x^{4}y^{6}}}{\sqrt{z}}\).
• Rationalizing: To make this expression rational (no square roots in the numerator), multiply both the numerator and the denominator by \(\sqrt{z}\). This operation does not change the value of the expression but transforms its form to \(x^{2}y^{3}\frac{\sqrt{6z}}{z}\).

This process ensures there are usually no radicals left in the numerator, thus producing a neat rational expression.

Variable Exponents

Variable exponents are important in understanding how to simplify and manipulate mathematical expressions, especially within roots.Here's the approach in context:

• Exponents Under Roots: When you have an even exponent, such as 4 or 6, it indicates a perfect square or higher power of the variable. For example, \(\sqrt{x^{4}} = x^{2}\) and \(\sqrt{y^{6}} = y^{3}\).
• Reducing Complexity: These properties enable us to simplify quite complex terms into straightforward ones, making rationalizing processes systematically manageable.

Understanding variable exponents aids in manipulating and simplifying algebraic expressions greatly, especially when dealing with powers and roots.