Problem 19 - Technical Mathematics with Calculus

Question

Multiplication of Radicals. Multiply and simplify. $$2 \sqrt{3} \text { by } 3 \sqrt{8}$$

Step-by-Step Solution

Answer

The product is \(12\sqrt{2}\).

Step 1: Multiply the Coefficients

First, multiply the coefficients (numbers outside the radical sign) together. In this case, multiply 2 by 3.

Step 2: Multiply the Radicands

Next, multiply the radicands (numbers inside the radical sign) together. Multiply \(\sqrt{3}\) by \(\sqrt{8}\).

Step 3: Simplify the Resulting Radicand

Simplify the radical \(\sqrt{24}\) by finding perfect square factors. \(24 = 4 \times 6\), and since \(\sqrt{4}\) is a perfect square, you can take it out of the radical and simplify.

Step 4: Multiply the Simplified Radicals

Multiply the coefficient obtained in Step 1 by the square root of the perfect square obtained in Step 3, and leave the remaining radicand under the radical sign.

Key Concepts

Simplifying Radical Expressions

When simplifying radical expressions, the goal is to make the expression as straightforward as possible. This often involves finding and removing perfect square factors from under the radical sign. To do this effectively, it helps to be familiar with the perfect squares up to at least 12 squared (which is 144).

Consider the example of multiplying and simplifying the expression \(2 \sqrt{3}\) by \(3 \sqrt{8}\). The first action is to treat the parts outside the radical (the coefficients) separately from the parts inside the radical (the radicands). After the initial multiplication of coefficients and radicands, you are left with a new radical expression that may not be fully simplified.

Take for instance \(\sqrt{24}\). This expression can be simplified because 24 contains a perfect square factor, in this case, 4. Therefore, \(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\). By extracting the square root of the perfect square (4), the radical becomes simpler and the multiplication can be completed.

Perfect Square Factors

Understanding perfect square factors can greatly simplify the process of manipulating radical expressions. A perfect square is a number that has an integer as its square root. In the multiplication of radicals, spotting these perfect squares will allow you to take them out of the radical for simplification.

For example, when simplifying \(\sqrt{24}\), you search for perfect square factors of 24. In this case, 4 and 6 are factors of 24, and 4 is a perfect square because \(2^2 = 4\). Once identified, \(\sqrt{24}\) can be split into \(\sqrt{4 \times 6}\), which in turn simplifies to \(2\sqrt{6}\). Knowing your perfect squares, such as 1, 4, 9, 16, up to 144 and possibly beyond, can save a lot of time in simplifying radical expressions.

Radical Arithmetic

Radical arithmetic involves operations with radical expressions, including addition, subtraction, multiplication, and division. In the case of multiplication, you first multiply the coefficients of the radicals and then multiply the radicands together.

Following our exercise example, starting with the multiplication of \(2 \sqrt{3}\) by \(3 \sqrt{8}\), the coefficients (2 and 3) are multiplied to get 6, and the radicands (\(\sqrt{3}\) and \(\sqrt{8}\)) are multiplied together to form \(\sqrt{24}\). This product of radicands can further be simplified as explained in the sections on simplifying radical expressions and perfect square factors.

In summary, to multiply radicals: multiply the coefficients, multiply the radicands, and then simplify the result by factoring out perfect squares. These steps are fundamental in performing radical arithmetic efficiently and accurately.