Problem 59 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\left(16 x^{8} y^{4}\right)^{3 / 4}\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \(8x^{6}y^{3}\)

Step 1: Factorize the given expression

First, break down the given expression, \(\left(16 x^{8} y^{4}\right)^{3 / 4}\), into simpler factors. The number 16 can be expressed as \(2^{4}\), \(x^{8}\) can be expressed as \(x^{2\*4}\) and \(y^{4}\) as \(y^{1\*4}\). So, the given expression can be written as \[\left(2^{4} x^{2*4} y^{1*4}\right)^{3 / 4}\]

Step 2: Apply Exponent Rule

According to the rule of exponents, when a power is raised to another power, we multiply the exponents. So, apply this rule and simplify each part of the expression.\[\left(2^{4\*3/4}\right) \left(x^{2\*4*3/4}\right) \left(y^{1\*4*3/4}\right)\]

Step 3: Simplify the expression

Now, calculate the exponents to simplify the expression.\[\left(2^{3}\right) \left(x^{6}\right) \left(y^{3}\right)\]Finally, we can see that the simplified expression is \[8x^{6}y^{3}\]

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions might seem tricky, but it is a key skill in algebra that helps solve problems more efficiently. By breaking down complex expressions into simpler forms, you can more easily understand and work with algebraic equations.
To simplify an expression like \( \left(16 x^{8} y^{4}\right)^{3 / 4} \), we first need to transform it into a format that is easier to manage. This often involves operations like factorization, which is the process of breaking numbers and variables down into their fundamental components.
By simplifying, we aim to rewrite the expression in the shortest and most manageable form possible while retaining its original meaning. This is especially useful for further calculations or solving equations where you need a clear and concise expression to work with.

• Break down the expression into manageable parts.
• Apply algebraic rules that simplify those parts.
• Combine the simplified elements to get a straightforward expression.

Simplified expressions are neat, clear, and elegant, allowing mathematicians to focus on solving problems rather than managing cumbersome equations.

Exponent Rules

Understanding exponent rules is crucial for simplifying expressions involving powers. Exponents tell us how many times to multiply a number by itself, and they follow specific rules when it comes to operations like multiplication or division.
One of the key exponent rules used in the exercise is the power of a power rule. This rule states that when you raise a power to another power, you need to multiply the exponents. For example, \((x^m)^n = x^{m\cdot n}\).
Let's see this rule in action with the expression \( \left(16 x^{8} y^{4}\right)^{3 / 4} \). First, break it down using the rule:

• For the coefficients and variables: \(2^{4}\), \(x^{8}\), and \(y^{4}\).
• Each is raised to \(\frac{3}{4}\), so multiply exponents: \(2^{4 \times \frac{3}{4}}\), \(x^{8 \times \frac{3}{4}}\), and \(y^{4 \times \frac{3}{4}}\).

This results in a simpler expression with new exponents which are easier to handle: \(2^3 x^6 y^3\). Knowing these rules helps significantly reduce complexity when dealing with large expressions.

Factorization

Factorization is all about taking apart an expression into its basic building blocks, making it easier to work with during simplification. When you factorize, you express numbers or expressions as products of simpler factors.
For instance, in the expression \( \left(16 x^{8} y^{4}\right)^{3 / 4} \), factorization plays a key role. Here's how it looks:

• Factor 16 as \(2^4\). This reduces the number's complexity when using exponent rules.
• Express \(x^8\) as \(x^{2\cdot 4}\) to align it with the root process.
• Express \(y^4\) as \(y^{1\cdot 4}\) for similar simplification purposes.

This step might seem minor, but it sets the stage for applying exponent rules more easily. By recognizing patterns and common factors, you can simplify expressions much quicker. Factorization is often the first and very necessary step in breaking down complex algebraic expressions into workable forms.