Problem 62 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{8 x^{5} y}{2 z}}\)

Step-by-Step Solution

Answer

\(\frac{2x^2 \sqrt{x yz}}{z}\)

Step 1: Simplify the Expression

To begin rationalizing the numerator, we first need to simplify the expression inside the square root by dividing inside the fraction. The given expression is \( \sqrt{\frac{8 x^5 y}{2 z}} \). Start by dividing \(8 x^5 y\) by \(2 z\):\[ \frac{8 x^5 y}{2 z} = \frac{8}{2} \cdot \frac{x^5 y}{z} = 4 \cdot \frac{x^5 y}{z} = \frac{4x^5y}{z} \]Now we have \(\sqrt{\frac{4x^5y}{z}}\).

Step 2: Rationalize the Numerator

We need to rationalize the numerator by multiplying the expression by \(\frac{\sqrt{z}}{\sqrt{z}}\). This involves:\[ \sqrt{\frac{4x^5 y}{z}} \times \frac{\sqrt{z}}{\sqrt{z}} = \frac{\sqrt{4x^5 y z}}{z} \]By doing this, the numerator becomes \(\sqrt{4x^5 yz}\).

Step 3: Simplify the New Expression

Take a closer look at the new expression to simplify it further if possible:The numerator is \(\sqrt{4x^5 yz}\). We can factor out the perfect squares:\[ \sqrt{4x^5 yz} = \sqrt{4} \cdot \sqrt{x^4} \cdot \sqrt{x} \cdot \sqrt{yz} = 2x^2 \sqrt{x yz} \]Thus, the expression becomes:\[ \frac{2x^2 \sqrt{x yz}}{z} \]

Key Concepts

Simplifying Expressions

When faced with a math problem, simplifying expressions is often the first crucial step. It means reducing a mathematical phrase to its simplest form, making it easier to understand and solve. In our exercise, we begin with the expression inside the square root: - First, divide each component of the numerator and denominator of the fraction. The original expression is \(\frac{8x^5y}{2z}\). - We divide each term: \(\frac{8}{2}\) simplifies to 4, and the fractions of variables inside \(\frac{x^5y}{z}\) remain as they are due to no direct simplification possible yet. At the end of simplifying, the complicated expression transforms into a more manageable form: \(\frac{4x^5y}{z}\). This new expression is shorter and guides us easily towards the further steps needed to solve it.

Square Roots

Understanding square roots is essential when solving expressions that contain them. A square root asks, "What number would I multiply by itself to get this number?" In our context, we are dealing with an expression under a square root symbol:- We encounter the square root of a fraction, \(\sqrt{\frac{4x^5y}{z}}\). Working with square roots and fractions involves both simplifying what's underneath the root sign and possibly outside it once rationalized.- Initially, to rationalize (or make the numerator a rational number), multiply both the numerator and denominator by the square root of \(z\), i.e., \(\sqrt{z}\), transforming our expression into \(\frac{\sqrt{4x^5yz}}{z}\).Square roots are simplified further into expressions that include perfect squares. Here, any perfect square inside \(\sqrt{4x^5yz}\) that can be extracted as a number out of the root helps in simplifying: such as \(\sqrt{4} = 2\) and \(\sqrt{x^4} = x^2\). These become coefficients that simplify our expression further: \(2x^2\sqrt{x yz}\).

Fraction Operations

Handling fraction operations means dealing with how fractions are derived, manipulated, and simplified in mathematical expressions. In our exercise, fraction operations play an integral role throughout:- Begin by tackling the initial fraction, \(\frac{8x^5y}{2z}\). By dividing the numbers directly and leaving the variables for later simplification, the expression is reduced to \(\frac{4x^5y}{z}\).- Once set under a square root, further operations involve multiplying by \(\frac{\sqrt{z}}{\sqrt{z}}\) to rationalize the numerator. This doesn't change the value, but rather transforms it for easier simplification.After rationalizing, the fraction \(\frac{\sqrt{4x^5yz}}{z}\) can undergo additional operations. By factoring and simplifying, we focus on multiplying and reducing terms both inside and outside the square root, finally arriving at \(\frac{2x^2\sqrt{x yz}}{z}\). This indicates all fraction operations involved ensure the expression is in a form ready for practical application or further arithmetic work.