Problem 63 - Algebra 2

Question

Solve \(-\sqrt[3]{x}+3=0\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -27\).

Step 1: Move '3' to the other side

The equation is \(-\sqrt[3]{x}+3=0\). Subtract '3' from both sides of the equation, this gives \(-\sqrt[3]{x} = -3\).

Step 2: Raise to power of three

In order to isolate \(x\), we raise both sides of the equation to power of three, obtaining \((-\sqrt[3]{x})^3 = (-3)^3\), simplifying this results in \(x = -27\).

Key Concepts

Cubic Roots

When we talk about cubic roots, we're discussing a special type of root in mathematics. The cubic root of a number is a value that, when multiplied by itself twice more, gives you the original number. For instance, the cubic root of 27 is 3, because when 3 is multiplied by itself three times (i.e., 3 x 3 x 3), it equals 27. In our exercise, you have \(-\sqrt[3]{x}\), which represents the cubic root of \(x\) being negative. This indicates we're looking for a number that, when cubed, results in a negative value—such as \-3\, whose cube is \(-27\).
This idea is crucial for understanding equations involving cubic roots, as these equations are based on the concept of finding what number, when cubed, will get you back to the original value. Cubic roots can be positive or negative, depending on the equation, and recognizing this helps determine potential solutions.

Algebraic Manipulation

Algebraic manipulation involves rearranging or simplifying equations to make them easier to solve. In our example, we begin with the equation \(-\sqrt[3]{x}+3=0\). The first step of the solution involves moving '3' from the left side of the equation to the right by subtracting 3 from both sides. This results in \(-\sqrt[3]{x} = -3\). Such movements across an equation constitute a key part of algebraic manipulation, as they help isolate the variable we want to solve for.
Algebraic manipulation is about making the equation simple to understand and work with, often turning it into a form where a solution is more clearly visible. You might use techniques like adding, subtracting, multiplying, or dividing both sides of the equation by the same number, which keeps the equation balanced while simplifying it.

Equation Solving Steps

Solving equations systematically involves a series of steps designed to isolate the variable and find its value. Here, the process starts with moving constants to the other side of the equation, helping to focus on the term with the variable. Once you simplify the equation as much as possible, the crucial step is to cube both sides. This action counteracts the cubic root and reveals the variable itself.
Here's a quick recap of the solving steps used:

• First, subtract 3 from both sides of the equation to remove additional terms and focus on the cubic root aspect.
• Then, raise both sides of the equation to the power of three, which is the opposite operation of taking a cubic root. This effectively eliminates the cubic root from one side of the equation.
• Finally, simplify the resulting expression to find the value of \(x\).

Following such structured solving steps ensures that you maintain balance in the equation while systematically arriving at the solution.