Problem 84 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ 3 \sqrt[3]{27}+12 \sqrt[3]{216} $$

Step-by-Step Solution

Answer

The simplified expression is 81.

Step 1: Find Cube Roots

First, we need to find the cube roots of the numbers inside the radicals. The cube root of 27 is 3 because \(3^3 = 27\). For 216, the cube root is 6 because \(6^3 = 216\). So we get: \[3 \sqrt[3]{27} = 3 \times 3 = 9\] \[12 \sqrt[3]{216} = 12 \times 6 = 72\]

Step 2: Add the Simplified Terms

Add the simplified terms from the previous step. We have: \[9 + 72\] Perform the addition to get the final answer.

Step 3: Simplify the Expression

Add the numbers together: \[9 + 72 = 81\] Thus, the simplified expression is 81.

Key Concepts

Cube Roots

Understanding cube roots is crucial when simplifying expressions involving radicals. A cube root asks for a number which, when multiplied by itself three times, will yield the original number. For instance, the cube root of 27 is found by determining which number, raised to the power of three, results in 27. In this example, 3 is that magic number since

• \(3 \times 3 \times 3 = 27\) or more formally,
• \(3^3 = 27\).

Similarly, for the number 216, the cube root is 6 because:

• \(6 \times 6 \times 6 = 216\), or
• \(6^3 = 216\).

Being able to find cube roots quickly is beneficial when simplifying radicals. Adopting techniques such as prime factorization can assist, allowing you to break down numbers into smaller, more manageable components.

Addition of Radicals

When adding radicals, simplification is your best friend. You must first ensure that the radicals are like terms—meaning they have the same index and radicand (the number or expression under the radical sign). Consider this similar to collecting like terms in algebra.In our problem, after simplifying each term, we ended up with two plain numbers without any radical left, which were:

• 9 from \(3 \sqrt[3]{27}\) and
• 72 from \(12 \sqrt[3]{216}\).

It is straightforward in this example since we transitioned from radicals to whole numbers, making the addition process simpler. This conversion made it possible to simply add the numbers directly:9 + 72 = 81.Ensuring the steps are properly followed guarantees that your final expression is at its simplest form.

Radical Expressions

Radical expressions involve terms with roots, such as square roots, cube roots, or any higher orders. These expressions can initially seem complex, which is why understanding the operations on them is essential. Radicals are typically represented using the radical sign (√), which indicates the root. When the root is not a square root, further clarification, such as a number, is written above and to the left of the symbol. Cube roots, like in our exercise, involved ‘3’ placed there. Simplifying a radical expression generally means making it easier to read by reducing it to its simplest form. Steps include:

• Identifying parts of the radical and solving or breaking them down,
• Converting radicals to whole numbers when possible, and
• Performing necessary additions or subtractions to further simplify the expression.

Mastering these skills allows you to handle various operations involving radicals, leading to more accurate results in your mathematical calculations.