Problem 66 - Elementary and 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

Examine the radicals in the expression: \(6 \sqrt[3]{5y}\) and \(3 \sqrt[3]{5y}\). In this case, both radicals are \(\sqrt[3]{5y}\), so they are 'like radicals' and can be combined.

Step 2: Combine Coefficients of Like Radicals

Since the radicals are like terms, you simply add their coefficients. Add the coefficients 6 and 3 together: \(6 + 3 = 9\).

Step 3: Write the Final Expression

After combining the coefficients, retain the common radical term: \(9 \sqrt[3]{5y}\). This is the simplified expression.

Key Concepts

Like Radicals

When working with radicals, understanding 'like radicals' is essential. Like radicals are expressions that have the same root and radicand. The radicand is the number or expression inside the radical. For example, let's consider \( \sqrt[3]{5y} \).

• The root here is the cube root, represented by the small 3.
• The radicand is \(5y\).

Like radicals allow us to work with them in a similar way to how we combine like terms in algebra. If two or more radicals have the exact same radicand and same root, they can be added or subtracted directly. In the exercise example, both radicals share the same \(\sqrt[3]{5y}\), making them like radicals.

Coefficients

Coefficients are the numerical factors that multiply a variable or a radical. In the expression \(6 \sqrt[3]{5y} + 3 \sqrt[3]{5y}\), the numbers \(6\) and \(3\) are the coefficients.

Here's how they function:

• Coefficients can be combined when they multiply like radicals.
• In our example, both terms have the same radical \(\sqrt[3]{5y}\), allowing us to add their coefficients.

We add \(6 + 3\) to get \(9\), maintaining the radical part \(\sqrt[3]{5y}\). So, the combined expression becomes \(9 \sqrt[3]{5y}\). By focusing on the coefficients, radical expressions can often be simplified significantly.

Cube Roots

Cube roots are specific types of roots where you find a value that, when multiplied by itself twice more, results in the original number or expression. The cube root of a number \(a\) is represented as \(\sqrt[3]{a}\).

Key points about cube roots:

• They are the opposite operation of cubing a number.
• Finding a cube root means determining a number which, when cubed, gives you back the original number.
• In cases involving variables, as seen in \(\sqrt[3]{5y}\), the operation involves the whole radicand \(5y\).

Working with cube roots, especially when the radicands are identical, allows simplification by combining terms, as seen in our exercise. Simplifying expressions with cube roots involves recognizing these patterns and applying the appropriate arithmetic operations on the coefficients.