Problem 65 - College Algebra

Question

Simplify the radical expressions in Exercises \(61-68\) $$\sqrt[3]{9} \cdot \sqrt[3]{6}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[3]{9} \cdot \sqrt[3]{6}\) is \(\sqrt[3]{54}\).

Step 1: Apply the Product Rule

Firstly, use the property of radicals that \(\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}\), and apply it in reverse. So the expression \(\sqrt[3]{9} \cdot \sqrt[3]{6}\) can be rewritten as \(\sqrt[3]{9 \cdot 6}\).

Step 2: Calculate the Product

Calculate the product, so the expression becomes \(\sqrt[3]{54}\).

Step 3: Simplify Cube Root

The cube root of \(54\) can't be simplified, as \(54\) isn't a perfect cube. Therefore, \(\sqrt[3]{54}\) is the final, simplified form of the expression.

Key Concepts

Cube Root

A cube root identifies a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, as

• 2 multiplied by itself three times (2 \( \times \) 2 \( \times \) 2) equals 8.

Cube roots are a bit different from square roots. They undo the cubing of a number. Thus, finding the cube root is like asking, "what number, when cubed, results in this number?".
When dealing with cube roots, especially in expressions like \( \sqrt[3]{9} \), you are looking for a number which repeated in a factor of 3 gives you 9. As 9 isn't a perfect cube, its cube root isn't a simple integer but remains in its radical form in many cases.

Product Rule for Radicals

The Product Rule for Radicals simplifies expressions with radicals by letting you multiply the terms under the radicals. It is expressed mathematically as \( \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab} \). This rule applies to cube roots, as well as any other radical expressions, as long as the roots are of the same degree.
In our exercise, we had \( \sqrt[3]{9} \times \sqrt[3]{6} \). By using the Product Rule for Radicals, this expression becomes \( \sqrt[3]{54} \) since 9 \( \times \) 6 equals 54. This method is useful because it often allows for further simplification and gives a unified radical expression. Apply it carefully to combine radicals under a single root while being mindful of keeping the index (in this case, 3, for cube roots) consistent.

Simplifying Radicals

Simplifying radicals is about making an expression as clean and understandable as possible. In many instances, this means reducing the number under the radical to its smallest possible factor form. However, it’s crucial to note that not all radicals can be simplified.
Taking our cube root example \( \sqrt[3]{54} \), it isn't a perfect cube and thus doesn't simplify to a nice, whole number. However, if the number under the radical was composed of smaller perfect cubes, we could break it down. In general, you can simplify cube roots by:

• Identifying perfect cube factors (e.g., \(2^3, 3^3 \)).
• Pulling these perfect cubes out of the radical.

If no perfect cube factors exist, the radical remains as is, signaling that further simplification isn't possible.