Problem 23 - Technical Mathematics with Calculus

Question

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

Step-by-Step Solution

Answer

6 \sqrt{3} \sqrt[3]{2}

Step 1: Distribute the Coefficients

Multiply the numerical coefficients outside the radical signs. In this case, multiply 3 by 2.

Step 2: Multiply the Radicals

Since the radicals have different indices (a square root and a cube root), they cannot be multiplied directly. The square root and cube root need to be handled separately.

Step 3: Rewrite the Expression if Necessary

If possible, the expression inside the radicals can be multiplied or expressed in a form that allows simplification. In this case, there is no simplification inside the radicals, so this step can be skipped.

Step 4: Combine the Results

Combine the results from Step 1 and Step 2, keeping the square root and cube root separate if they cannot be simplified further. In this case, write the answer as the product of the coefficients and the separate radicals.

Key Concepts

Radical Expressions

When we talk about radical expressions, we're referring to mathematical expressions that include a root, such as the square root or the cube root. The general form of a radical expression is \( \sqrt[n]{x} \), where \( n \) is the index of the root and \( x \) is the radicand—the number under the radical sign.

Working with radical expressions can involve several operations, including addition, subtraction, multiplication, and division. To multiply radicals, a key understanding is that \( \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b} \), but this only holds true when the indices of the radicals are the same.

When faced with a radical expression involving different indices, such as the textbook exercise \(3 \sqrt{3}\) multiplying \(2 \sqrt[3]{2}\), the process involves additional steps because you can't combine different types of roots directly. This brings an interesting mix of radical rules and ensures students gain a robust understanding of how to manipulate and simplify various radical expressions.

Simplifying Radicals

Simplifying radicals is a process used to rewrite radical expressions in their simplest form. This process can make calculations easier and expressions more comprehensible. There are several strategies to simplify radicals:

• Identifying and extracting perfect powers from under the radical sign.
• Breaking down the radicand into prime factors.
• Rationalizing the denominator when dealing with fractions.

Interestingly, not all radical expressions can be simplified—sometimes you've already reached the simplest form. In the textbook problem, you'll notice that \(3 \sqrt{3}\) and \(2 \sqrt[3]{2}\) cannot be simplified through multiplication as they have different indices. This is an excellent example of a scenario where understanding the boundaries of simplification is just as important as knowing how to simplify.

Distributive Property of Multiplication

One of the fundamental properties of multiplication is the distributive property. This property comes in handy when you have to multiply a single term by a sum or difference of terms, expressed as \(a(b + c) = ab + ac\). But how does it apply to radicals?

In the case of multiplying radicals, this property is reflected when you multiply the coefficients (the numbers outside the radical signs) and then deal with the radicals separately. Returning to our exercise, Step 1 applied this principle: \(3 \times 2 = 6\), which is the multiplication of the coefficients outside the \(\sqrt{3}\) and \(\sqrt[3]{2}\).

The distributive property reminds us that multiplication is commutative; we can reorganize and group numbers for ease of calculation without changing the result. When you face an operation involving radicals with different indices, as in our example, understanding this property helps to structure the approach: multiplying coefficients and radicals separately to reach the final expression.