Problem 55 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. $$ 20 \sqrt[3]{4}-15 \sqrt[3]{4} $$

Step-by-Step Solution

Answer

The simplified form is \(5 \sqrt[3]{4}\).

Step 1: Identify Like Radicals

We are given the expression \(20 \sqrt[3]{4} - 15 \sqrt[3]{4}\). Both terms contain the same cube root \(\sqrt[3]{4}\), which means they are 'like radicals'. Like radicals can be combined just like like terms in algebra.

Step 2: Combine Like Radicals

Since the radicals are like terms, you can combine them by subtracting the coefficients of the radicals. The coefficients are 20 and 15, so perform the subtraction: \(20 - 15 = 5\).

Step 3: Write the Simplified Expression

After combining the coefficients, rewrite the expression with the common radical: The simplified form is \(5 \sqrt[3]{4}\).

Key Concepts

Cube Roots

A cube root is a special type of radical expression. While a square root seeks a number which, when multiplied by itself twice gives a certain value, a cube root finds a number that when used three times in a multiplication gives the original value. For example, the cube root of 8 is 2 because

• 2 \( \times \) 2 \( \times \) 2 = 8

Cube roots can be identified by the radical sign \( \sqrt[3]{} \), with the 3 indicating it's a cube root. If you're working with cube roots, you're often dealing with expressions that have this form. In the problem we examined, both terms share the same cube root: \( \sqrt[3]{4} \). This similarity is key because it makes these terms 'like radicals', which means they can be combined easily. Understanding cube roots helps simplify complex expressions, transforming difficult parts into manageable elements.

Combining Like Terms

Combining like terms is a fundamental concept in algebra that involves simplifying expressions. Like terms are terms that contain the same variable raised to the same power, or in the case of radicals, the same root expression. For example, \( 20 \sqrt[3]{4} \) and \( -15 \sqrt[3]{4} \) are like terms because both include the cube root \( \sqrt[3]{4} \). This allows for simplification by combining their coefficients:

• Perform the arithmetic operation on the coefficients: \( 20 - 15 \) results in 5.
• Remember, you can only combine like terms (or like radicals) this way.

By combining them, you're effectively reducing the expression into a simpler form.

Coefficients in Algebra

In algebra, coefficients are numerical or constant factors in terms involving variables or roots. When you look at \( 20 \sqrt[3]{4} \), 20 is the coefficient of the radical \( \sqrt[3]{4} \). Coefficients are crucial as they denote how many times a term is used in an addition or a subtraction process.

• They act like multipliers for the root expressions.
• Understanding coefficients helps in effectively combining like terms.

In the simplification process, knowing how to handle coefficients allows you to focus on the arithmetic while keeping the radical part consistent. In our example, subtracting the coefficients (20 and 15) ultimately yields 5, forming the simplified expression \( 5 \sqrt[3]{4} \). This demonstrates the power of recognizing and manipulating coefficients in algebraic simplification.