Question

Solve the equation. Check your solution $$ 2 \sqrt[3]{x}-13=-5 $$

Short answer

The solution is \(x = 64 \).

Step-by-step solution

Isolate the cubic root term

To isolate the cubic root term, add 13 to both sides of the equation: \(2 \sqrt[3]{x}-13+13=-5+13\), which simplifies to \(2 \sqrt[3]{x}=8\)

Remove the cubic root

To remove the cubic root, first eliminate the scalar 2 by dividing both sides by 2 : \( \sqrt[3]{x}=4 \). Then, cube both sides of the equation: \( (\sqrt[3]{x})^3 = (4)^3 \), which simplifies to \( x=64 \)

Check your solution

To check the solution, substitute 64 into original equation \(2 \sqrt[3]{64}-13=-5\), which simplifies to \(-5 = -5\), thus validating our solution. Thus, \(x = 64\) is indeed solution to the original equation

Key concepts

Isolate Cubic Root Term

Grasping algebra can be challenging, but understanding each step enhances your mathematical prowess. Let's begin with isolating the cubic root term, an essential step in solving cubic root equations.
Consider the equation from our exercise:
\[\begin{equation}2 \sqrt[3]{x}-13=-5\end{equation}\]
Initially, our primary objective is to separate the cubic root term from other elements. To isolate it, simply perform basic arithmetic operations. In this given equation, we add 13 to both sides, balancing the equation to make it \[\begin{equation}2 \sqrt[3]{x}=8\end{equation}\]
This step is crucial, as it simplifies the equation and prepares it for the next phase, where we tackle the cubic root directly.

Removing Cubic Roots in Equations

With the cubic root term isolated, our next logical step is removing the cubic root to find the value of 'x'. This part is akin to unwrapping a present — the process reveals what's inside.
Here's how we do it: we have \[\begin{equation}2 \sqrt[3]{x}=8\end{equation}\]
Our equation now has a lone cubic root term, albeit with a multiplier. The first order of business is to divide by that multiplier— in this case, 2—to yield \[\begin{equation}\sqrt[3]{x}=4\end{equation}\]
Now it's time for the grand reveal: we cube both sides of the equation. This act cancels out the cube root on the left and amplifies the right side, giving us \[\begin{equation} x=64\end{equation}\]
By cubing both the root and the number, we've essentially freed 'x' from its cubic constraints. It's straightforward yet powerful and forms the core of solving such equations.

Checking Algebraic Solutions

Just as a mathematician checks their calculations, we must verify our solution to ensure accuracy. This is the quintessence of solving equations: validation.
Checking the solution protects against false positives—a critical step in algebra. For our example, we have determined \[\begin{equation} x=64\end{equation}\]
To check, we substitute 'x' with 64 in the original equation: \[\begin{equation}2 \sqrt[3]{64}-13=-5\end{equation}\]
Upon simplification, both sides of the equation remain equal, affirming that \[\begin{equation} -5 = -5\end{equation}\]
Thus, our solution stands verified. This gratifying step not only reinforces our confidence but also strengthens our understanding of the equation's dynamics. Always make it a habit to check your solutions—your surety in math will thank you.