Problem 9 - Elementary and Intermediate Algebra

Question

Multiply. $$ \sqrt[3]{3} \sqrt[3]{2} $$

Step-by-Step Solution

Answer

\(\root[3]{6}\).

Step 1: Understand the problem

The goal is to multiply two cube roots: \(\root[3]{3} \times \root[3]{2}\).

Step 2: Use the property of cube roots

Recall the property of cube roots that states: \(\root[3]{a} \times \root[3]{b} = \root[3]{a \times b}\).

Step 3: Apply the property

Apply the property to combine the cube roots: \(\root[3]{3} \times \root[3]{2} = \root[3]{3 \times 2}\).

Step 4: Simplify the expression

Multiply the numbers inside the cube root: \(\root[3]{3 \times 2} = \root[3]{6}\).

Key Concepts

cube roots

Cube roots help us find a number that, when multiplied by itself 3 times, gives the original number. For example, the cube root of 8 is 2, since 2 × 2 × 2 = 8. In mathematical notation, the cube root of a number 𝑥 is written as \(\root[3]{𝑥}\). Understanding cube roots is crucial for problems related to volume and geometry, as well as for multiplication of radicals.
Cube roots are similar to square roots but instead of squaring, we work with cubes. This makes them very useful for solving equations where variables are raised to the third power.
In the exercise, we have two cube roots: \(\root[3]{3}\) and \(\root[3]{2}\). We will be using their properties to simplify and multiply them.

properties of roots

The properties of roots, especially cube roots, make many mathematical operations easier. A key property we use here is:
\(\root[3]{a} \times \root[3]{b} = \root[3]{a \times b}\).
This is because roots and exponents are closely related. With cube roots, we are dealing with the exponent of \(1/3\).
Understanding this property allows us to combine and simplify expressions involving cube roots.
For example, combining \(\root[3]{3}\) and \(\root[3]{2}\) using this property gives us \(\root[3]{3 \times 2}\).
This simplification is helpful in reducing complex expressions to simpler forms.

multiplication of radicals

Multiplying radicals, such as cube roots, can be straightforward if we remember the key properties. As we saw in the step-by-step solution, multiplying \(\root[3]{3}\) and \(\root[3]{2}\) uses the rule \(\root[3]{a} \times \root[3]{b} = \root[3]{a \times b}\).
Let's break it down further:

• Step 1: Identify the radicals you are multiplying: \(\root[3]{3}\) and \(\root[3]{2}\).


• Step 2: Apply the property: \(\root[3]{3} \times \root[3]{2} = \root[3]{3 \times 2}\).


• Step 3: Simplify the multiplication inside the radical: \(\root[3]{6}\).

By understanding and using these properties, you can greatly simplify complex roots and make them easier to manage in various mathematical problems.