Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\left(x^{6} x^{3}\right)^{1 / 3}\)

Step-by-Step Solution

Answer

The expression \(\left(x^{6} x^{3}\right)^{1 / 3}\) simplifies to \(x^{3}\).

Step 1: Apply product of powers property

Using the product of powers property, \(x^{m} x^{n} = x^{m+n}\), \(\left(x^{6} x^{3}\right)^{1 / 3} = x^{6+3} = x^{9}\). So the expression simplifies to \((x^{9})^{1 / 3}\).

Step 2: Apply power of a power property

Using the power of a power property, \((x^{m})^{n} = x^{m \times n}\), the expression further simplifies as \((x^{9})^{1 / 3} = x^{9 \times 1 / 3} = x^{3}\).

Key Concepts

Product of Powers

One of the fundamentals of working with exponents is the product of powers property. This property simplifies the multiplication of like bases raised to different exponents.

Here's how it works: when you multiply two exponents with the same base, you add their powers. For example, with base 'x', if you have

• \(x^m \times x^n\),
• you simply perform \(x^{m+n}\).

In the original exercise, this rule was used to combine \(x^6\) and \(x^3\) into \(x^{6+3} = x^{9}\).

This simplifies our initial terms significantly, making it easier to manage expressions and solve the problems at hand without overwhelming calculations.

Power of a Power

The power of a power property comes into play when we deal with exponents raised to another exponent.

The rule is quite straightforward: when one power is raised to another, you multiply the exponents. So, when you encounter

• \((x^m)^n\),
• you compute it as \(x^{m \times n}\).

In the exercise, after applying the product of powers, we ended up with \((x^9)^{1/3}\).

Utilizing the power of a power property, it became \(x^{9 \times 1/3} = x^3\).

This approach helps to streamline expressions, making them more manageable and easier to interpret.

Simplifying Expressions

Simplifying expressions is about rewriting them in their simplest form for easier comprehension and computation.

Using properties like product of powers and power of a power makes the task straightforward and efficient. Initially, expressions might look complicated, but by methodically applying these properties:

• Product of powers: combine like bases.
• Power of a power: simplify nested exponents.

In our exercise, we simplified \((x^6 x^3)^{1/3}\) to \((x^9)^{1/3}\) using these techniques.

Ultimately, the final expression reduces to \(x^3\).

This simplification not only aids in clearer understanding but also assists in further mathematical operations or solving more complex problems.