Problem 26 - Algebra and Trigonometry

Question

Simplify the expression. \(\sqrt[3]{54}-\sqrt[3]{16}\)

Step-by-Step Solution

Answer

The simplified expression is \(\sqrt[3]{2}\).

Step 1: Breakdown Cube Roots

We start by identifying the cube roots in the expression: \(\sqrt[3]{54}\) and \(\sqrt[3]{16}\). Our goal is to simplify these cube roots if possible.

Step 2: Prime Factorization

Next, we perform prime factorization for both numbers under the cube roots. - For 54, the prime factors are \(54 = 2 \times 3^3\).- For 16, the prime factors are \(16 = 2^4\), which is \(2 \times 2^3\).

Step 3: Simplify Each Term

Now we simplify each cube root by matching cubes from their prime factorizations:- \(\sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = 3 \times \sqrt[3]{2}\).- \(\sqrt[3]{16} = \sqrt[3]{2^4} = 2 \times \sqrt[3]{2}\).

Step 4: Substitute Back Into Expression

We substitute the simplified cube roots back into the original expression:\(\sqrt[3]{54} - \sqrt[3]{16} = (3 \times \sqrt[3]{2}) - (2 \times \sqrt[3]{2})\).

Step 5: Combine Like Terms

Finally, we combine like terms by subtracting the coefficients of \(\sqrt[3]{2}\):\((3 - 2) \times \sqrt[3]{2} = 1 \times \sqrt[3]{2} = \sqrt[3]{2}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks—called prime numbers—which cannot be divided further, except by one or themselves. When simplifying expressions involving cube roots, prime factorization helps us identify cube terms and simplify them.
Let's look at an example from our original exercise: to simplify \( \sqrt[3]{54} \), we first find its prime factors.
- 54 can be expressed as \( 2 \times 3^3 \). Here, 3 is raised to the power of 3, allowing us to take it out of the cube root.
Similarly, for \( \sqrt[3]{16} \), prime factorization gives us \( 16 = 2^4 = 2 \times 2^3 \). This means we can also simplify it by taking \( 2^3 \) out as a 2, while another \( 2 \) stays inside the cube root.
Using prime factorization makes it easier to see which factors can be simplified when under a cube root. This is the first step towards simplifying such algebraic expressions.

Combining Like Terms

Combining like terms is a fundamental process in algebra that involves merging terms with the same variable or base. This concept becomes especially important when simplifying expressions.
In the provided exercise, after we apply prime factorization, we rewrite the cube roots of our numbers into forms that can be combined. We ended up with:

• \( 3 \times \sqrt[3]{2} \) from \( \sqrt[3]{54} \)
• \( 2 \times \sqrt[3]{2} \) from \( \sqrt[3]{16} \)

Both of these have the common component \( \sqrt[3]{2} \), allowing them to be combined.
We perform simple subtraction on the coefficients (3 and 2): - \( (3 - 2) \times \sqrt[3]{2} = 1 \times \sqrt[3]{2} = \sqrt[3]{2} \).
Thus, combining like terms effectively simplifies the expression to its final form.

Algebraic Expression Simplification

Algebraic expression simplification involves reducing expressions to their simplest form by performing operations such as factoring, distributing, and combining like terms. It's essential for solving equations efficiently and understanding underlying mathematical relationships.
In this specific exercise, the simplification involved several interconnected steps:

• Breaking down cube roots through prime factorization.
• Simplifying the terms outside of the cube roots.
• Combining like terms to achieve the simplest form of the expression.

The final simplified form of our expression, \( \sqrt[3]{54} - \sqrt[3]{16} \), became \( \sqrt[3]{2} \) after these procedures. Each of these operations is critical because they reveal that, despite the initial complexity, many expressions can be reduced to more manageable forms.
Mastering these techniques makes tackling similar algebraic problems in the future much more intuitive and straightforward.