Problem 22 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt[3]{2} \cdot \sqrt[3]{4} $$

Step-by-Step Solution

Answer

The simplest form is 2.

Step 1: Apply the Product Property of Radicals

Recall that when multiplying like radicals, you can combine them under a single radical. For cube roots, this means that \( \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} \). In this exercise, apply this property to combine \( \sqrt[3]{2} \) and \( \sqrt[3]{4} \), resulting in \( \sqrt[3]{2 \times 4} \).

Step 2: Simplify the Multiplication in the Radicand

Multiply the numbers under the cube root. Here, \( 2 \times 4 = 8 \). Thus, the expression becomes \( \sqrt[3]{8} \).

Step 3: Find the Cube Root

The next step is to simplify \( \sqrt[3]{8} \). Since \( 2^3 = 8 \), the cube root of \( 8 \) is \( 2 \). Therefore, \( \sqrt[3]{8} = 2 \).

Step 4: Conclusion

After simplifying, the product \( \sqrt[3]{2} \cdot \sqrt[3]{4} \) simplifies to \( 2 \). Therefore, the simplest form of the given expression is \( 2 \).

Key Concepts

Product Property of Radicals

The Product Property of Radicals is an important rule that helps in multiplying radicals. Simply put, this property allows you to combine two radicals of the same order into one. For instance, when dealing with cube roots, you can multiply the numbers inside the cube root together and then find the cube root of that product. Here's the formula:

• \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \)

For the exercise at hand, this property let us combine \( \sqrt[3]{2} \) and \( \sqrt[3]{4} \) into a single radical expression: \( \sqrt[3]{2 \times 4} \). This process simplifies your work, saving you from separately dealing with each root. Remember, this only works when the radicals are of the same degree, like both being cube roots.
Using this property makes the initial step in radical multiplication straightforward and sets the groundwork for further simplification.

Simplification of Radicals

Simplification means taking a complex expression and making it easier to understand. For radicals, simplification often involves finding perfect powers and reducing the expression inside the radical sign. In our exercise, we started with \( \sqrt[3]{2 \times 4} \), which then simplifies through multiplication as \( \sqrt[3]{8} \).
The idea is to complete any arithmetic under the radical first. This means multiplying the numbers under the cube root without finding their cube roots right away. Once we have a simpler form like \( \sqrt[3]{8} \), it can be reduced further by recognizing that 8 is a perfect cube (since \( 2^3 = 8 \)). In practice, always look for opportunities to replace the radicand with simpler numbers.

• Calculate what's inside the radical first.
• Look for and apply any perfect powers within the radicand.

These steps ensure that by the time you take the radical, it's as straightforward as possible, making the final result clear and easy to compute.

Multiplication under Radicand

Multiplication under the radicand refers to performing arithmetic operations within the radical sign. In a cube root situation, such as \( \sqrt[3]{2 \times 4} \), we multiply the numbers 2 and 4 under the radicand first. This simplifies to \( \sqrt[3]{8} \).
Doing calculations under the radicand helps bring the expression into a manageable format. It allows for a direct solution without the need to simplify later. Remember, the order of operations still applies - handle multiplication before addressing properties of the cube roots.

• Ensure all multiplications are completed under the radicand before finding the root.
• Always check for opportunities where the result can be simplified using properties of exponents.

This method clarifies the expression step-by-step, ensuring that when you take the cube root, you have the cleanest form of the expression possible, which is crucial for arriving at the correct and simplest solution.