Problem 78 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. $$ 3 \sqrt[3]{24}+\sqrt[3]{81} $$

Step-by-Step Solution

Answer

The simplified form of \(3\sqrt[3]{24}+\sqrt[3]{81}\) is \(9\sqrt[3]{3}\)

Step 1: Simplify 3 \sqrt[3]{24}

First, let's simplify \(3 \sqrt[3]{24}\). We recognize that 24 can be factored into \(2^3\) and 3. Therefore, we rewrite the term as \(3 \sqrt[3]{2^3 \cdot 3}\). The cube root of \(2^3\) is 2, so the term simplifies to \(6\sqrt[3]{3}\).

Step 2: Simplify \(\sqrt[3]{81}\)

Next, let's simplify \(\sqrt[3]{81}\). By observing, we find that 81 can be factored into \(3^4\). Therefore, we rewrite the term as \(\sqrt[3]{(3^3) \cdot 3}\). The cube root of \(3^3\) is 3, so the term simplifies to \(3\sqrt[3]{3}\).

Step 3: Add Simplified Terms

Now that both terms are simplified, they can be added together because they have the same radical. Therefore, the expression is simplified to \((6\sqrt[3]{3}) + (3\sqrt[3]{3}) = 9\sqrt[3]{3}\).

Key Concepts

Simplifying radicals

When dealing with the expression \(3 \sqrt[3]{24} + \sqrt[3]{81}\), one must simplify the cube roots. This process involves breaking down numbers under the radical sign into their prime factors.
Particularly for cube roots, we need to identify factors that are perfect cubes. For example, in the number 24, which can be expressed as \(2^3 \times 3\), the \(2^3\) is a perfect cube that can be taken out of the radical, simplifying \(\sqrt[3]{24}\) to \(2\sqrt[3]{3}\).
Simplifying radicals helps in handling the terms easily, especially when they need to be added or subtracted.

Factoring

Factoring involves breaking down a number into its prime factors. This is a vital step in simplifying cube roots.
For example, to factor 24, we identify it as \(2^3 \times 3\). This allows us to pull \(2^3\) out of the cube root.
Similarly, when factoring 81, we find it to be \(3^4\), which can be expressed as \(3^3 \times 3\), helping in simplifying \(\sqrt[3]{81}\). Factoring helps reveal hidden perfect cubes and is a key step in simplifying cube root expressions.

Adding like terms

Once cube roots are simplified, like terms can be added together. Terms are considered 'like' if they have the same radical component.
In our example, both simplified terms are \(6\sqrt[3]{3}\) and \(3\sqrt[3]{3}\), sharing \(\sqrt[3]{3}\) under the cube root symbol.
Simply combine the coefficients by addition: \(6 + 3\), leading to the simplification \(9\sqrt[3]{3}\).
Adding like terms allows expressions to be condensed into simpler forms for easier handling.

Cube root simplification

Cube root simplification involves reducing numbers under the cube root sign by extracting any perfect cubes.
For \(\sqrt[3]{24}\), recognize \(2^3\) as a perfect cube, which simplifies \(\sqrt[3]{24}\) to \(2\sqrt[3]{3}\). Similarly, for \(\sqrt[3]{81}\), we take \(3^3\) outside the radical, resulting in \(3\sqrt[3]{3}\).
Simplification helps in performing operations like addition or subtraction, providing a clearer, simpler result.