Problem 95 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( ab^2 \).

Step 1: Introduce the Problem

The expression we need to simplify is \( \left(a^{3} b^{6}\right)^{1/3} \). We will rewrite this expression with rational exponents while simplifying it.

Step 2: Apply the Power Rule

Using the power rule in exponents, \((x^m)^n = x^{m \cdot n}\), we apply this to both terms in the expression. For \(a^3\), it's \( (a^3)^{1/3} = a^{3 \cdot 1/3} = a^{1} \). For \(b^6\), it's \((b^6)^{1/3} = b^{6 \cdot 1/3} = b^{2}\).

Step 3: Combine the Results

Now combine the simplified terms. We have \(a^{1} = a\) and \(b^2\). So the expression \( \left(a^{3} b^{6}\right)^{1/3} \) simplifies to \(ab^2\).

Key Concepts

Simplifying Expressions

Simplifying expressions often makes mathematical equations much easier to work with. In the context of algebra, simplification generally refers to reducing complex expressions into more manageable and straightforward forms without changing their value.
By rewriting expressions in a simplified manner, calculations and subsequent operations become less prone to error and quicker to complete.
Consider the expression \(\left(a^{3} b^{6}\right)^{1/3}\). The goal here is to simplify it so that it is easier to interpret or use in further mathematical tasks.
Some essential aspects of simplification include:

• Identifying patterns or rules that can be applied, such as exponent rules.
• Rewriting a given expression by applying these rules systematically.
• Ensuring that the simplified expression is equivalent to the original one.

By addressing these elements, we ensure that simplification is effectively reducing the complexity of an expression.

Power Rule in Exponents

The power rule for exponents is a fundamental tool in algebra for manipulating expressions with powers. It states that when you raise a power to another power, you simply multiply the exponents.
Mathematically, this can be represented as \((x^m)^n = x^{m \cdot n}\).
When applied in the context of rational exponents, this rule allows us to transform expressions like \(\left(a^{3} b^{6}\right)^{1/3}\) into a simpler form.
Here's how it works:

• Apply the power rule separately on each term in the expression.
• For \(a^3\), use: \( (a^3)^{1/3} = a^{3 \cdot 1/3} = a^{1} = a\).
• For \(b^6\), use: \( (b^6)^{1/3} = b^{6 \cdot 1/3} = b^{2}\).

By understanding and applying the power rule, you can simplify expressions efficiently and accurately, ensuring you can work more easily with complex algebraic forms.

Expression Simplification Steps

Simplifying an expression requires careful attention to detail and a good grasp of algebraic rules. The process typically involves several steps to systematically reduce the expression. Let's break it down:

• Identify the components: Look closely at the expression \(\left(a^{3} b^{6}\right)^{1/3}\) to identify each term and their exponents.
• Apply relevant rules: Use rules like the power rule for exponents. Here, you apply the rule to each term: \((a^3)^{1/3}\) and \((b^6)^{1/3}\).
• Simplify each term: Multiply the exponents for each term. Calculate \(a^{1/3 \cdot 3} = a^1 = a\) and \(b^{1/3 \cdot 6} = b^2\).
• Combine the results: Once each term is simplified, combine them for the final result: \(ab^2\).

Thus, the initial expression simplifies to \(ab^2\), making it more concise and simpler to use in further calculations.