Problem 96 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \left(64 \mathrm{x}^{3} \mathrm{y}^{18}\right)^{1 / 6} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x^{1/2}y^3 \).

Step 1: Understand the Expression

We begin with the expression \( (64x^3y^{18})^{1/6} \). Our goal is to simplify the expression and express it using rational exponents.

Step 2: Distribute the Exponent

We apply the exponent \( \frac{1}{6} \) to each part of the product inside the parentheses: \( (64)^{1/6} \), \( (x^3)^{1/6} \), and \( (y^{18})^{1/6} \).

Step 3: Simplify \( (64)^{1/6} \)

Since \( 64 = 2^6 \), applying the \( \frac{1}{6} \) power gives us \( 64^{1/6} = (2^6)^{1/6} = 2 \).

Step 4: Simplify \( (x^3)^{1/6} \)

We can use the property of exponents \( (a^m)^n = a^{m \cdot n} \). Applying this gives us \( (x^3)^{1/6} = x^{3 \cdot \frac{1}{6}} = x^{1/2} \).

Step 5: Simplify \( (y^{18})^{1/6} \)

Using the same exponent rule, \( (y^{18})^{1/6} = y^{18 \cdot \frac{1}{6}} = y^3 \).

Step 6: Combine the Results

Now, we combine the results from each part: \( 2x^{1/2}y^3 \). This is our simplified expression using rational exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex expressions into more manageable forms. For the given expression \((64x^3y^{18})^{1/6}\), we want to make it easier to understand and work with. This often means reducing expressions by following a logical set of steps.In our exercise, we start by looking at the expression as a whole and recognize it can be simplified by distributing the outer exponent to each factor inside the parentheses. This step is crucial in simplification because it allows us to deal with each part individually.

When simplifying, always remember:

• Look for common patterns. Here, the base of the number 64 can be seen as \(2^6\).
• Apply properties of exponents to break down the expression.
• Simplify each component separately before combining results.

This process of simplification can greatly enhance our understanding of the expression and make further computations straightforward.

Properties of Exponents

The properties of exponents are powerful tools that help us manipulate and simplify expressions involving exponential terms. These rules apply regardless of whether we're dealing with integers, fractions, or variables. Here are some key properties we use:**- The Power of a Power Rule** This rule states \((a^m)^n = a^{m imes n}\). It's used for simplifying terms where an exponent is raised by another exponent. In our exercise, we use it to simplify \((x^3)^{1/6}\) to \(x^{1/2}\) and \((y^{18})^{1/6}\) to \(y^3\).

**- Product Rule and Quotient Rule**While not used directly here, these rules are also important. The product rule, \(a^m imes a^n = a^{m+n}\), and the quotient rule, \(\frac{a^m}{a^n} = a^{m-n}\), help simplify expressions by combining or reducing like terms.

Understanding and applying these properties help transform complex expressions into simpler versions that are easier to evaluate or solve.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They're the foundation of algebra and are used to represent real-world problems mathematically. In our context, the expression \((64x^3y^{18})^{1/6}\) is an algebraic expression that includes constants like 64, variables like \(x\) and \(y\), and employ operations like exponentiation.Expressions can come in various forms, and simplifying them often involves several steps. We often use algebraic expressions to model situations and solve problems. Key things to remember are:

• Pay attention to the structure of the expression—recognize coefficients, variables, and exponents.
• Use consistent methods, such as distribution and the properties of exponents, to simplify expressions.
• Always perform operations step-by-step to avoid mistakes.

This structured approach will help make sense of the components of any expression and manipulate them effectively, paving the way to solve equations and interpret mathematical models with confidence.