Problem 66 - Intermediate Algebra

Question

Simplify by combining like radicals. All variables represent positive real numbers. $$ 6 \sqrt[3]{5 y}+3 \sqrt[3]{5 y} $$

Step-by-Step Solution

Answer

The simplified expression is \(9 \sqrt[3]{5y}\).

Step 1: Identify Like Radicals

In the given expression \(6 \sqrt[3]{5y} + 3 \sqrt[3]{5y}\), we first need to identify the 'like radicals.' Like radicals have the same radicand (the expression under the radical) and the same index. Both terms have the cube root of \(5y\), thus they are like radicals: \(\sqrt[3]{5y}\).

Step 2: Combine Like Radicals

Since we have determined that the radicals are like, we can combine them by adding their coefficients. The coefficients are the numbers in front of the radicals. For \(6 \sqrt[3]{5y}\), the coefficient is 6, and for \(3 \sqrt[3]{5y}\), the coefficient is 3. Add the coefficients together: \(6 + 3 = 9\).

Step 3: Write the Simplified Expression

After adding the coefficients, multiply the result by the common radical to write the simplified expression. So, \(9\sqrt[3]{5y}\) is the simplified form of the original expression.

Key Concepts

Combining Like Terms

When working with radicals, a crucial step is to recognize when you can combine like terms. Like terms in radical expressions have the same radicand and the same index of the root. For instance, in the expression \(6 \sqrt[3]{5y} + 3 \sqrt[3]{5y}\), both terms share the same cube root \(\sqrt[3]{5y}\). This means they are like radicals.

To combine these, focus on the coefficients—the numbers that are multiplying the radicals. In this exercise, the coefficients are 6 and 3. By adding these coefficients together, we get 9. Therefore, we can simplify the expression to \(9 \sqrt[3]{5y}\). This simplification works just like combining like terms in algebra, where you consolidate terms with the same variables by summing their coefficients.

In summary, identify like radicals by looking for the same radicand and index, and then add their coefficients together to simplify the expression. This concept of combining like terms makes working with radicals much more manageable.

Radical Expressions

Radical expressions are any expressions that include a radical symbol, which indicates the root of a number or expression. The expression under the radical is called the radicand. For example, in \(\sqrt[3]{5y}\), the radicand is \(5y\) and the index of the root is 3, indicating a cube root. This expression is a cube root radical expression.

• The index of the radical tells you what type of root you’re finding. An index of 2 implies a square root, while an index of 3 implies a cube root, and so on.
• The radicand can be a number, a variable, or a combination of both, as seen in \(5y\).

To work with radical expressions, understand that you can manipulate them by applying rules similar to regular numbers. For instance, like terms can be added or subtracted if their radicals match. You can also multiply and divide radical expressions, keeping track of the rules for radicals.

Learning how to handle radical expressions is essential for solving more complex mathematical problems where roots play a crucial role.

Cube Roots

Cube roots are a type of radical expression where the index is 3. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, since \(2 \times 2 \times 2 = 8\). In mathematical notation, a cube root is represented by \(\sqrt[3]{x}\), where \(x\) is the radicand.

• Cube roots are important in various fields including geometry, physics, and finance, where understanding volume or scaling is necessary.
• To simplify cube root expressions, focus on simplifying the radicand where possible, which often involves recognizing perfect cubes or factoring.

In the expression \(9 \sqrt[3]{5y}\), the cube root indicates a relationship where each component of the radicand is considered three times over—they are, in a sense, repetitive multipliers. Understanding cube roots enhances your numerical fluency and helps when solving equations where straightforward multiplication might not reveal the answer directly.

By mastering cube roots, you'll be better equipped to tackle a wide array of mathematical problems, introducing concepts of greater depth and complexity.