Problem 20 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(4 \sqrt[3]{24}-6 \sqrt[3]{3}+13 \sqrt[3]{81}\)

Step-by-Step Solution

Answer

The simplified expression is \(41 \sqrt[3]{3}\).

Step 1: Factor Inside the Radicals

We start by factoring inside each radical to see if there are any perfect cubes. For \( \sqrt[3]{24} \), we have \( 24 = 2^3 imes 3 \), so \( \sqrt[3]{24} = \sqrt[3]{2^3 \times 3} = 2 \sqrt[3]{3} \).For \( \sqrt[3]{81} \), we have \( 81 = 3^4 = 3^3 \times 3 = (3^3) \times 3 \), so \( \sqrt[3]{81} = \sqrt[3]{3^4} = \sqrt[3]{3^3 \times 3} = 3 \sqrt[3]{3} \).

Step 2: Use Distributive Property

Now, substitute the factored forms of the radicals back into the expression:\[4 \sqrt[3]{24} = 4 \times 2 \sqrt[3]{3} = 8 \sqrt[3]{3}\]\[-6 \sqrt[3]{3}\]\[13 \sqrt[3]{81} = 13 \times 3 \sqrt[3]{3} = 39 \sqrt[3]{3}\]

Step 3: Simplify Expression by Combining Like Terms

Combine all like terms containing \( \sqrt[3]{3} \):\[8 \sqrt[3]{3} - 6 \sqrt[3]{3} + 39 \sqrt[3]{3} = (8 - 6 + 39) \sqrt[3]{3} = 41 \sqrt[3]{3}\]

Step 4: Final Result

The expression \(4 \sqrt[3]{24} - 6 \sqrt[3]{3} + 13 \sqrt[3]{81}\) simplifies to \(41 \sqrt[3]{3}\).

Key Concepts

Simplifying Radicals

Simplifying radicals is a process of breaking down the expression under the radical sign so that it can be expressed in the simplest form possible. Radicals often involve finding square roots, cube roots, or similar roots. Here, we will focus on cube roots. To simplify a radical like \( \sqrt[3]{24} \), you should look for perfect cube factors. A perfect cube is a number that can be written as another whole number raised to the power of three.

For instance:

• First, factor the number 24 into its prime factors: \(24 = 2^3 \times 3\).
• Next, break down the cube root: \( \sqrt[3]{24} = \sqrt[3]{2^3 \times 3}\. \)

This simplifies to \(2 \sqrt[3]{3}\) because \(2^3\) is a perfect cube, and \( \sqrt[3]{2^3} = 2\). Simplifying radicals will make it much easier to combine terms later.

Perfect Cubes

A perfect cube is any number that can be expressed as the cube of an integer. For example, 8 is a perfect cube because it can be written as \(2^3\). Recognizing perfect cubes is a key step in simplifying radicals, especially when dealing with expressions like \( \sqrt[3]{81} \).

To break down \( 81 \), we can factor it as \( 3^4 = 3^3 \times 3 \). Here, \(3^3\) is a perfect cube because it equals 27, which is \(3\) raised to the power of 3. Thus, \( \sqrt[3]{81} = \sqrt[3]{3^4} = 3 \sqrt[3]{3} \).

Recognizing these perfect cubes allows for the simplification process to continue efficiently. Identifying and utilizing the perfect cubes reduces complex terms, simplifying the expression and preparing it for combining like terms.

Combining Like Terms

Combining like terms is the final step in simplifying expressions after radicals have been simplified. It involves adding or subtracting coefficients of terms with the same radical component. For the given expression \( 4 \sqrt[3]{24} - 6 \sqrt[3]{3} + 13 \sqrt[3]{81} \), the terms are rewritten (based on previous simplifications) as \( 8 \sqrt[3]{3} - 6 \sqrt[3]{3} + 39 \sqrt[3]{3} \).

To combine them:

• Identify the common radical part, which in this case is \( \sqrt[3]{3} \).
• Add or subtract the coefficients: \( 8 - 6 + 39 = 41 \).
• Keep the common radical with the new coefficient: \( 41 \sqrt[3]{3} \).

It's important to ensure that only like terms, those with identical radical parts, are combined. This process results in a more concise and simplified expression, making calculations and further algebraic manipulations much easier.