Problem 50 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \left(32^{1 / 5} x^{2 / 3}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(32^{3/5} \cdot x^2\).

Step 1: Apply Power of a Power Rule

The first step is to apply the power of a power rule, which states that when you raise a power to another power, you multiply the exponents. Our expression is \( \left(32^{1/5} \cdot x^{2/3}\right)^3 \). Applying this rule, we get:\[32^{(1/5) \cdot 3} \cdot x^{(2/3) \cdot 3}\]

Step 2: Simplify the Exponents

Now, simplify the exponents by multiplying out the fractions:- For \(32\): \( (1/5) \cdot 3 = 3/5 \), so we have \(32^{3/5}\).- For \(x\): \( (2/3) \cdot 3 = 2 \), so we have \(x^2\).Therefore, the simplified expression is:\[32^{3/5} \cdot x^2\]

Step 3: Write with Positive Exponents

Our expression is already written with positive exponents, as required. The expression is fully simplified as:\[32^{3/5} \cdot x^2\]

Key Concepts

Power of a Power Rule

When you're dealing with exponents, the 'power of a power rule' is a handy property to remember. If you have an expression with an exponent raised to another exponent, you simply multiply the two exponents together. This makes complicated expressions easier to handle and simplifies calculations.

For example, if you have the expression \((a^m)^n\), the power of a power rule allows you to combine the exponents, resulting in \(a^{m \cdot n}\). In our exercise, we applied this rule to \( (32^{1/5} \cdot x^{2/3})^3 \). By multiplying the exponents inside the brackets by 3, we simplified it to \(32^{3/5} \cdot x^2\).

Applying this rule correctly is crucial for simplifying any expression involving exponents efficiently and accurately.

Simplifying Exponents

Simplifying exponents involves reducing the expression to its simplest form while respecting the rules of exponents. In our exercise, once the power of a power rule was applied, we multiplied the exponents to simplify the terms. Let's explain this in detail:

• For the base 32, the exponent \(\frac{1}{5}\) is multiplied by 3 to give \(\frac{3}{5}\). Thus, the expression becomes \(32^{3/5}\).
• For the variable \(x\), the exponent \(\frac{2}{3}\) is multiplied by 3, resulting in \(x^2\).

By simplifying these exponents, you ensure that the expression is easy to read and further calculations can be easily managed.

Always remember, the goal of simplifying is to make the expression more approachable while preserving its value and form. This is essential when dealing with complex mathematical problems.

Positive Exponents

A central requirement in rearranging expressions is to ensure all exponents are positive. Using positive exponents makes expressions straightforward to interpret and further simplifies calculations.

In mathematical expressions, negative exponents indicate reciprocals. For instance, \(x^{-a}\) equates to \(\frac{1}{x^a}\). However, such expressions can lead to complications, especially in calculus and algebra, so converting them to positive exponents is favorable.

In our exercise, both \(32^{3/5}\) and \(x^2\) are already positive, making the task simpler. Maintaining positive exponents avoids potential confusion, especially in advanced math, and cleans up your work for presentation or future manipulations.

A clear, positive, and simplified expression is always easier to work with or apply to future calculations, ensuring clarity and precision in mathematical communication.