Problem 56 - Intermediate Algebra

Question

Simplify by combining like radicals. $$ 30 \sqrt[3]{6}-10 \sqrt[3]{6} $$

Step-by-Step Solution

Answer

The simplified expression is \(20 \sqrt[3]{6}\).

Step 1: Identify Like Radicals

The expression involves two cube roots, \(30 \sqrt[3]{6}\) and \(-10 \sqrt[3]{6}\). Both terms have the same radical part, \(\sqrt[3]{6}\), making them like radicals.

Step 2: Combine Coefficients

Since the radicals are the same, we can combine the terms by adding their coefficients. Calculate the new coefficient by subtracting: \[(30 - 10) \sqrt[3]{6} = 20\sqrt[3]{6}.\]

Step 3: Simplified Expression

The expression simplifies to \(20 \sqrt[3]{6}\) after combining the like radicals.

Key Concepts

Combining Like Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, and so forth. When we talk about combining like radicals, what we mean is that we are looking for radical terms within an expression that have the same index and radicand (the number inside the radical). Like the way one can combine like terms in algebra, like radicals can also be combined based on their coefficients.

• The index of a radical is the small number outside the radical sign that denotes the type of root, like the 3 in cube roots.
• The radicand is the number inside the radical, like the 6 in \(\sqrt[3]{6}\).

In the original problem, both terms indeed share the same index (they are cube roots) and the same radicand (6). This qualifies them as like radicals, allowing them to be simplified by combining coefficients. It's important to remember that you can only combine radicals that are of the same type - just as you can't combine \(2x\) and \(2y\), you can't combine \(\sqrt[3]{6}\) and \(\sqrt[3]{12}\) directly.

Cube Roots

Cube roots are a specific type of radical expression where the index of the root is 3. Let's explore cube roots further to understand how they fit within radical expressions. When you see \(\sqrt[3]{x}\), it refers to a number which, when multiplied by itself three times, results in \(x\).

• The expression \(\sqrt[3]{8} = 2\) because \(2 \times 2 \times 2 = 8\).
• Cube roots can deal with both positive and negative numbers, unlike square roots which are undefined for negative radicands without introducing complex numbers.

Understanding cube roots is essential because they often appear in various mathematical contexts. They are crucial when solving equations involving cubics, and simplifying radical expressions with cube roots, as we've seen in the original exercise, is a common task. The key takeaway on cube roots is identifying them with their index of 3 and recognizing when they can be combined with like terms.

Coefficients in Radical Expressions

Coefficients in radical expressions act similarly to coefficients in algebraic expressions. They are the numbers placed in front of radicals and signify how many times that particular radical is to be counted. For example, in the expression \(30 \sqrt[3]{6}\), 30 is the coefficient.To simplify expressions by combining like radicals, the following steps are useful:

• Identify the coefficients of like radicals. In \(30 \sqrt[3]{6} - 10 \sqrt[3]{6}\), the coefficients are 30 and -10.
• Combine these coefficients through standard arithmetic operations (i.e., addition or subtraction). Here, we subtract to get 20, as \(30 - 10 = 20\).
• Multiply the resulting coefficient with the shared radical. Thus, transforming \(30 \sqrt[3]{6} - 10 \sqrt[3]{6}\) to \(20 \sqrt[3]{6}\).

By focusing on coefficients, it becomes straightforward to simplify radical expressions. This method ensures clarity and accuracy in calculations, making working with radicals much less daunting.