Question

Multiplication of Radicals. Multiply and simplify. $$3 \sqrt{2} \text { by } 2 \sqrt[3]{3}$$

Short answer

The simplified expression is \(6 \sqrt{2} \sqrt[3]{3}\). Since the radicals are different types, they cannot be combined further.

Step-by-step solution

Distribute the Coefficients

Multiply the coefficients (numbers outside the radicals) together: 3 times 2.

Combine the Radicals

For the radicals, since they are not the same type (one is a square root and the other is a cube root), you cannot multiply the inside numbers directly. The multiplication of different types of radicals does not simplify directly.

Write the Final Expression

Multiply the results from step 1 and write them together with the unsimplified radical part.

Key concepts

Simplify Radicals

The process of simplifying radicals involves making the expression as concise and straightforward as possible. Simplifying a radical means to factor out any perfect squares from under the square root (or perfect cubes from under the cube root), and then bringing them in front of the radical sign as their corresponding integer values. For example, simplifying \( \sqrt{18} \) would involve recognizing that 18 is 9 times 2, and 9 is a perfect square. Thus, \( \sqrt{18} = \sqrt{9\times2} = 3\sqrt{2} \).

When it comes to the multiplication of different types of radicals as in the exercise \( 3 \sqrt{2} \text{ by } 2 \sqrt[3]{3} \), it's important to know that you cannot directly simplify across different types of roots. In this case, a square root and a cube root are involved. They each address different dimensions of a shape's volume or area, which means they can't be combined into a single radical term. As a result, the radical part of the expression remains unsimplified, but we can still multiply any coefficients involved.

Radical Expressions

A radical expression is an expression that involves a root, such as a square root or cube root. The symbol for the square root is \( \sqrt{\cdot} \) and for the cube root, it's \( \sqrt[3]{\cdot} \). Besides the square root and cube root, there are higher roots like fourth roots, fifth roots, and so on. The number inside the radical sign is called the radicand.

In our exercise, the expressions \( 3\sqrt{2} \) and \( 2\sqrt[3]{3} \) are both radical expressions, with the first involving a square root and the second involving a cube root. In mathematics, it's crucial to understand various properties of radicals to perform operations like multiplication, division, addition, or subtraction. However, while coefficients (the numbers in front of the radicals) can be multiplied easily, the radicands must obey specific rules and cannot always be directly combined, especially if they involve different kinds of roots.

Distributing Coefficients

When multiplying radical expressions, distributing coefficients is the first step. This operation is similar to the distributive property used in algebra where you multiply terms outside a parenthesis with terms inside. For radicals, distributing coefficients means multiplying the numbers located outside of the radical signs first before dealing with the radicals themselves.

In our given problem, we multiply the coefficients \(3 \) and \(2 \) first to get \(3 \times 2 = 6\). This multiplication is straightforward because we are dealing with the non-radical part of the expressions. Remember, regardless of the types of radicals involved, the coefficients will always distribute across the multiplication as they are not under any radical sign. It's only after this step that we consider the radicands and the type of roots in the expression. In this case, since the radical types differ (square root versus cube root), we are unable to combine the radicals, and they remain as they are after multiplying the coefficients.