Question

What is the solution(s) for \(x\) in this equation? $$ \frac{12}{\sqrt[3]{x}}=4 $$ (A) \(-3\) and 81 (B) \(-27\) and 27 (C) 9 only (D) 27 only

Short answer

The solution to the equation is 27 only, corresponding to option (D).

Step-by-step solution

Isolate the Cube Root

To begin solving the equation \( \frac{12}{\sqrt[3]{x}} = 4 \), we must first isolate the cube root term. Multiply both sides by \( \sqrt[3]{x} \) to get \( 12 = 4\sqrt[3]{x} \).

Divide Both Sides by 4

Now, divide both sides of the equation \( 12 = 4\sqrt[3]{x} \) by 4 to solve for the cube root: \( 3 = \sqrt[3]{x} \).

Solve for x by Cubing

To eliminate the cube root, cube both sides of \( 3 = \sqrt[3]{x} \): \( 3^3 = x \). This simplifies to \( 27 = x \).

Verify Solutions with Options

With \( x = 27 \), compare to the given options. Only option (D), 27 only, matches our solution.

Key concepts

Solving Equations

When we talk about solving equations, we're referring to finding the value(s) that make the equation true. In this exercise, our key task is to solve the equation \( \frac{12}{\sqrt[3]{x}} = 4 \). The goal is to isolate the variable, which is often done step-by-step. By

• Isolating the term with the variable, which in this case is the cube root \( \sqrt[3]{x} \)
• Manipulating the equation to simplify or remove complex expressions like fractions or cube roots
• Verifying that the solution fits within the context or constraints of the problem

we ensure that the solution is correct. Always keep the equation balanced by performing the same operations on both sides. This ensures that we properly solve for the variable without altering the equality of the equation. Understanding these principles is crucial for successfully solving algebraic equations.

Cube Root

The concept of a cube root is central to this problem because it appears in the term \( \sqrt[3]{x} \). A cube root of a number \( y \) is a number \( x \) such that \( x^3 = y \). In simpler terms, when you cube \( x \), you get the original number \( y \).

In the given equation \( \frac{12}{\sqrt[3]{x}} = 4 \), isolating the cube root involves rewriting the equation to express \( \sqrt[3]{x} \) on its own. Once isolated, like with \( 3 = \sqrt[3]{x} \), you can cube both sides to eliminate the cube root:

• Cubing \( 3 \) gives \( 3^3 = 27 \)
• This implies \( x = 27 \)

Remember, eliminating cube roots by cubing is an essential skill in algebra, and practicing this will help solidify your understanding of how cube roots function in equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables or express them in a more convenient form. In this exercise, we see several techniques at play:

• Multiplying both sides: In the original equation, multiplying both sides by \( \sqrt[3]{x} \) helps remove the fraction, giving us the more manageable equation \( 12 = 4\sqrt[3]{x} \).
• Dividing: By dividing both sides by 4, we simplify the expression further, isolating the cube root to yield \( 3 = \sqrt[3]{x} \).
• Cubing: Cubing both sides allows us to remove the cube root entirely, solving for \( x \).

Each of these moves requires a solid understanding of balancing equations and executing precise operations to ensure no mistakes. Practicing these algebraic manipulations is crucial for solving more complex problems in mathematics. With these foundational skills, you can manipulate any form of equation you encounter in algebra.