Problem 5 - Intermediate Algebra

Question

Use the product rule to multiply. See Example \(I\). \(\sqrt[3]{4} \cdot \sqrt[3]{9}\)

Step-by-Step Solution

Answer

The product is \(\sqrt[3]{36}\); it is already simplified.

Step 1: Understanding the Problem

We have the expression \(\sqrt[3]{4} \cdot \sqrt[3]{9}\). Our task is to multiply these two cube roots using the product rule for radicals, which states that \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}\) if \(n\) is the same for both radicals.

Step 2: Apply the Product Rule

According to the product rule for radicals, we can multiply the expressions inside the roots. Therefore, we use \(\sqrt[3]{4} \cdot \sqrt[3]{9} = \sqrt[3]{4 \cdot 9}\). Simplifying the multiplication inside the cube root: \(4 \cdot 9 = 36\). So, the expression is now \(\sqrt[3]{36}\).

Step 3: Simplify the Expression

To simplify \(\sqrt[3]{36}\), we need to determine if 36 has a perfect cube factor. The prime factorization of 36 is \(36 = 2^2 \times 3^2\). This doesn't directly give a perfect cube, so \(\sqrt[3]{36}\) is the simplified form since 36 itself isn't a perfect cube.

Key Concepts

Cube Roots

Understanding cube roots is crucial when working with expressions involving radicals. A cube root of a number is another number that, when multiplied by itself three times, results in the original number. It's denoted as \( \sqrt[3]{x} \). So, for example, \( \sqrt[3]{8} = 2 \) because 2 multiplied by itself three times results in 8.

• Cube roots are closely related to cube numbers, like 1, 8, 27, 64, etc.
• Not every number has a whole number as its cube root.

When working with cube roots in mathematical expressions, it's essential to recognize if the number inside the cube root has any perfect cube factors. This understanding allows for simplification, which can make solving and understanding expressions more manageable. Cube roots provide a foundation for more advanced topics in algebra and are a stepping stone to mastering mathematical operations involving radicals.

Radicals

Radicals are expressions that include roots, such as square roots or cube roots. The symbol \( \sqrt{ } \) represents a radical, and the small number to the left (if present) is what type of root it is.

• Square root: \( \sqrt{4} = 2 \)
• Cube root is shown as \( \sqrt[3]{27} = 3 \)

The general form for a radical is \( \sqrt[n]{x} \), where \( n \) is the degree of the root. If \( n \) is not shown, it’s typically a square root. Radicals are necessary for simplifying expressions and solving equations that involve powers and roots.
It's important when dealing with radicals to remember their relationship with exponents. Roots can be converted into fractional exponents, which can often make them easier to manipulate in algebraic operations. Understanding radicals is a key skill in algebra that helps in simplifying expressions and solving complex equations more intuitively.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more concise, clearer, or standard form. When working with radicals, simplifying often means reducing the expression inside the root as much as possible.

• Identifying and factoring out perfect powers, such as perfect squares or cubes, makes simplification easier.
• Use the properties of radicals: \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab} \).

For instance, with the expression \( \sqrt[3]{36} \), we simplify by checking if 36 can be broken down into a multiplication of a perfect cube. Although 36 itself isn't a perfect cube, such simplification can make further operations more manageable.
Simplification is crucial because it reduces the complexity of problems. It helps to lay the groundwork for more advanced mathematical manipulations and ensures results are expressed as simply as possible, making them easier to interpret and use in real-world applications. Always strive to simplify where you can for a clearer mathematical solution.