Problem 80 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \sqrt[3]{\frac{54 x^{2} y^{4}}{2 x^{5} y}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3y}{x}\).

Step 1: Simplify the Fraction Inside the Radicand

First, write the expression inside the cube root as a single fraction. The expression is \[ \frac{54 x^{2} y^{4}}{2 x^{5} y}. \] To simplify it, divide the coefficients and use the laws of exponents to simplify the powers of variables. So, divide 54 by 2 to get 27. For the variables: - Subtract the exponents of \(x\) using \(x^{a}/x^{b} = x^{a-b}\), giving \(x^{2-5} = x^{-3}\).- Subtract the exponents of \(y\) using \(y^{a}/y^{b} = y^{a-b}\), giving \(y^{4-1} = y^{3}\).Thus, the simplified expression inside the radicand is \[ 27 x^{-3} y^{3}. \]

Step 2: Apply the Cube Root to Each Factor

Next, apply the cube root to each part of the expression separately. Firstly, the cube root of 27: \[ \sqrt[3]{27} = 3. \]Secondly, use the property that \( \sqrt[3]{a^b} = a^{b/3} \). Apply this to each variable:- \( \sqrt[3]{x^{-3}} = x^{-3/3} = x^{-1} \).- \( \sqrt[3]{y^{3}} = y^{3/3} = y^{1} \).Combine all parts together, so the expression becomes \[ 3 x^{-1} y^{1}. \]

Step 3: Simplify the Expression with Positive Exponents

Finally, express the expression using only positive exponents. The factor \(x^{-1}\) is the same as \(\frac{1}{x}\), hence, \[ 3 x^{-1} y^{1} = 3 \frac{y}{x}. \] This is the simplified expression with rational exponents.

Key Concepts

Cube Root Simplification

Simplifying a cube root means breaking down an expression involving a cube root into its simplest form. The original exercise involves a cube root:

• Expression inside the cube root: \( \frac{54 x^{2} y^{4}}{2 x^{5} y} \).

The first step in cube root simplification is to simplify the fraction inside the cube root. This involves:

• Dividing the numerical coefficients: \( \frac{54}{2} = 27 \).
• Applying the laws of exponents to any variables (we'll discuss this more later).

Once simplified, the cube root can be applied to each term separately. For numbers like 27, this results in a cleaner, simpler number, specifically \( \sqrt[3]{27} = 3 \). For variables, we use rational exponents, converting the cube root into a power of 1/3. This allows us to simplify powers of variables more systematically.

Laws of Exponents

Understanding the laws of exponents is crucial. They allow us to simplify expressions with variables raised to powers. When dealing with a fraction inside a cube root, as seen here, we're often required to subtract the exponents when dividing similar terms. Here's what we do:

• For the variable \(x\), you subtract the exponents: \(x^{2} / x^{5}\) becomes \(x^{2-5} = x^{-3}\).
• For the variable \(y\), the subtraction yields \(y^{4} / y^{1} = y^{4-1} = y^{3}\).

These properties help convert complex expressions into simpler forms. Then, when applying a cube root, every variable exponent \(a^b\) inside the root is expressed as \(a^{b/3}\), as laws of exponents allow converting radicals to fractional powers. This method lays the foundation for dealing with more advanced algebraic expressions efficiently.

Positive Exponents

Expressing exponents as positive is a final step in simplification, and it's applicable when you end up with negative exponents. From our simplified form in the exercise, \(3 x^{-1} y^{1}\), the task requires us to eliminate negative exponents. This is accomplished by:

• Realizing that \(x^{-1}\) is equivalent to \(\frac{1}{x}\).

Substitute these parts to ensure all exponents are positive. This turns the expression into \(3 \frac{y}{x}\). This approach is important because it aligns with standard mathematical practice of expressing solutions in their most conventional form, making them clearer and easier to understand. It's all about transforming terms to be as simple and direct as possible.