Problem 28 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[3]{2}+\sqrt[3]{16} $$

Step-by-Step Solution

Answer

The expression simplifies to \(3\sqrt[3]{2}\).

Step 1: Simplify Each Term Separately

Start by simplifying each cube root term in the expression. For \( \sqrt[3]{2} \), since 2 is a prime number that is less than any cubes, it cannot be simplified further. For \( \sqrt[3]{16} \), notice that 16 can be expressed as \( 2^4 \).

Step 2: Simplify \( \sqrt[3]{16} \)

We saw that 16 is \(2^4\). Since we are dealing with a cube root, \( \sqrt[3]{2^4} = (2^4)^{1/3} \). Simplify this by using exponent rules: \( (2^4)^{1/3} = 2^{4/3} \). This can be split into \( 2^{1} \cdot 2^{1/3} \). Thus, \( \sqrt[3]{16} = 2\sqrt[3]{2} \).

Step 3: Combine Like Terms

We now have the expression \( \sqrt[3]{2} + 2\sqrt[3]{2} \). Since these terms have the same cube root factor \( \sqrt[3]{2} \), they can be combined like algebraic terms: \( 1\sqrt[3]{2} + 2\sqrt[3]{2} = 3\sqrt[3]{2} \).

Key Concepts

Cube Roots

Cube roots are a specific type of radical expression where a number, known as the radicand, is raised to the 1/3 power. The cube root of a number is a value that, when multiplied by itself three times, gives the radicand again. In mathematical notation, the cube root of a number 'a' is written as \( \sqrt[3]{a} \).
For example, the cube root of 8 is 2, because when you multiply 2 by itself three times, you get 8 (i.e., \( 2 \times 2 \times 2 = 8 \)). When dealing with cube roots, especially those involving complex radicands, identifying perfect cubes can simplify expressions considerably.

• To find cube roots: Look for values that are multiplied by themselves three times to get the number inside the radical.
• If the radicand is not a perfect cube, simplify as much as possible.

Exponent Rules

Understanding exponent rules is crucial for simplifying expressions involving radicals or cube roots. These rules help you manipulate expressions containing powers or roots. One of the most used rules involves multiplying exponents, such as \( (x^a)^b = x^{ab} \). This tells us that when raising a power to another power, you multiply the exponents.
In our main expression, \( \sqrt[3]{16} \) can be reframed as \( 16 = 2^4 \), leading to \( (2^4)^{1/3} \). According to exponent rules, this simplifies to \( 2^{4/3} \). Breaking this down further allows the expression \( 2^4 = 2^3 \times 2^1 \). This methodology states that \( 2^{4/3} = 2^1 \times 2^{1/3} \), showcasing how exponents can simplify complex calculations.

• Power of a power: Multiply the exponents together.
• Product of powers: When bases are the same, add the exponents.
• Division of powers: When dividing, subtract exponents of like bases.

These rules form a toolkit for unraveling complex algebraic expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. These expressions are the building blocks of most mathematical calculations.
When simplifying a radical expression like \( \sqrt[3]{2} + \sqrt[3]{16} \), recognizing common terms is important. Just like in polynomial addition, terms with the same radical can be combined.
Here, \( \sqrt[3]{2} \) is the common term in both parts of the expression. By identifying that \( \sqrt[3]{16} = 2 \sqrt[3]{2} \), it becomes evident that you can treat each cube root as an algebraic term, similar to how algebraic expressions are managed. Combining these like terms is straightforward: one \( \sqrt[3]{2} \) plus two \( \sqrt[3]{2} \) yields three \( \sqrt[3]{2} \).

• Identify like terms: Look for expressions with the same variable part.
• Combine coefficients: Add the numbers in front of the like terms.

Understanding this helps in simplifying terms and making expressions more manageable.