Problem 12 - Elementary and Intermediate Algebra

Question

Perform each operation, if possible. $$ \text { a. } 5+6 \sqrt[3]{6} $$ $$ \text { b. } 5(6 \sqrt[3]{6}) $$ $$ \text { c. } \frac{30 \sqrt[3]{15}}{5} $$ $$ \text { d. } \frac{\sqrt[3]{15}}{5} $$

Step-by-Step Solution

Answer

a. No simplification possible. b. Result is \(30 \sqrt[3]{6}\). c. Simplifies to \(6 \sqrt[3]{15}\). d. No further simplification.

Step 1: Simplify Expression for Part a

In Part (a), we have the expression \(5+6 \sqrt[3]{6}\). This expression is already in its simplest form because the terms \(5\) and \(6 \sqrt[3]{6}\) are not like terms and cannot be combined. The operation to perform is simply writing the expression as it is.

Step 2: Perform Multiplication for Part b

For Part (b), we need to multiply the number 5 with the term \(6 \sqrt[3]{6}\). First, multiply the constant 5 by 6, giving \(30 \). Thus, the resulting expression after performing the operation is \(30 \sqrt[3]{6}\).

Step 3: Simplify Division in Part c

In Part (c), we divide \(30 \sqrt[3]{15}\) by 5. First, divide 30 by 5, which yields 6. The expression simplifies to \(6 \sqrt[3]{15}\), as the cube root remains unchanged by division of the coefficients.

Step 4: Simplify Expression in Part d

Part (d) involves dividing \(\sqrt[3]{15}\) by 5. This expression cannot be simplified further because the cube root \(\sqrt[3]{15}\) and the number 5 are not like terms, and the division does not result in simplification. Therefore, the expression remains \(\frac{\sqrt[3]{15}}{5}\).

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions is a key operation that makes complex equations and terms easier to work with. When simplifying, the goal is to reduce an expression to its simplest form, while keeping its original value intact.
This involves performing operations like addition, subtraction, multiplication, and division, as well as using properties of radicals or exponents. One of the most important aspects to remember during simplification is to combine like terms when possible. Like terms are terms that contain the same variables raised to the same power. In the case of radical expressions, like terms may include numbers that can be combined under the same radical. For example, in the expression \[5 + 6 \sqrt[3]{6}\]it cannot be simplified further because the terms aren't like terms: 5 is a whole number, whereas \(6 \sqrt[3]{6}\) involves a cube root.

Cube Roots

Cube roots, denoted as \(\sqrt[3]{\cdot}\), play a significant role in algebra by helping us understand and simplify equations involving power of three. A cube root of a number is a value that when multiplied by itself twice results in the original number. For instance, the cube root of 8 is 2, because multiplying 2 three times (2 × 2 × 2) gives us 8.
This operation is crucial when simplifying expressions that involve radicals raised to another power.When dealing with cube roots in algebraic expressions, it's essential to discern if the cube roots can be simplified by factoring the radicand— the number under the root.
Sometimes, prime factorization can be helpful to identify any numbers that can be pulled out of the cube root, similar to how we simplify square roots.

Like Terms

Recognizing and combining like terms is a fundamental skill to master in algebra. Like terms are terms in an expression that have identical variable parts, and they can be combined through addition or subtraction.For example, in the expression \(3x + 4x\),the terms can be combined to \(7x\),since both terms contain the variable \(x\) raised to the same power.
Similarly, when dealing with radicals, like terms are those that involve the same radical part.Within the scope of an expression like \(5 + 6 \sqrt[3]{6}\), the terms are not alike because \(5\) is a standalone whole number and \(6 \sqrt[3]{6}\) contains a cube root.
Understanding this concept is critical, as incorrectly combining unlike terms is a common mistake. Under normal algebraic rules, they must be treated separately unless they share the same variable or radical component.