x*x*x is equal to? Find Solution

To solve the equation x*x*x is equal to $a$, where a is a given constant, we are looking for all possible values of x that satisfy this cubic equation. The process involves finding the roots of the cubic equation, which can be real or complex numbers depending on the value of a. Below is a detailed discussion on how to solve this equation, including the case when a=0 and when $a\mathrm{\ne }0$.

Case 1: a $=0$

When a=0, the equation simplifies to:

${\mathrm{x*x*x\; is\; equal\; to\; 0,\; or\; x}}^3=0$

The only solution to this equation is $x=0$, as $0^3=0$. This is a straightforward case with a single real root.

Case 2: a $\mathrm{\ne }0$

When a$\mathrm{\ne }0$, solving$x^3=\mathrm{a\; or\; x*x*x\; is\; equal}$  to 2 becomes more interesting. We can find the roots using different methods, including algebraic manipulation, the use of the cubic formula, or numerical methods. However, for simplicity and direct application, we will focus on the algebraic method here.

Real Solution

Every non-zero a value in the equation x*x*x is equal to $a$ has exactly one real root, which can be found using the cube root function. The real root is given by:

$x=\sqrt[3]{}$a​

This formula provides the principal cube root of a. If a is positive, this root is also positive; if a is negative, the root is negative.

Complex Solutions

In addition to the real root, there are also two complex roots for each non-zero a, which are not as straightforward to calculate without delving into complex numbers. These roots arise from the property that a cubic equation has three roots (counting multiplicity) according to the Fundamental Theorem of Algebra.

For a general a, the complex roots can be found using the formula that involves Euler’s formula or trigonometric considerations, but they are typically represented as:

$x=\sqrt[3]{a}\cdot (\cos \frac{2\mathrm{<span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="minner"><span\; class="mord"><span\; class="mfrac"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist"><span\; class="sizing\; reset-size6\; size3\; mtight"><span\; class="mord\; mtight"><span\; class="mord\; mathnormal\; mtight">\pi k</span></span></span></span></span></span></span></span></span></span></span>}}{3}+i\sin \frac{2\mathrm{<span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="minner"><span\; class="mord"><span\; class="mfrac"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist"><span\; class="sizing\; reset-size6\; size3\; mtight"><span\; class="mord\; mtight"><span\; class="mord\; mathnormal\; mtight">\pi k</span></span></span></span></span></span></span></span></span></span></span>}}{3}),k=1,2$

Where i is the imaginary unit (i.e., $i^2=-1$), and k takes on the values 1 and 2 to give the two distinct complex roots. This representation shows that the complex roots are equally spaced around the circle in the complex plane, centered at the origin with a radius equal to the cube root of the absolute value of a.

Numerical Example

Let’s solve a specific equation as an example, say x${}^3=8$ or x*x*x is equal to 8.

1. Real Root: The real root is x =
   $\sqrt[3]{8}=2$ 
2. Complex Roots: Although we won’t calculate these in detail here, they would follow the form provided above and represent the two other solutions in the complex plane.

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Conclusion

Solving

$x^3=\mathrm{a\; or\; x*x*x\; is\; equal\; to\; a}$ involves finding one real root directly through the cube root of a and, if needed, calculating two additional complex roots for a comprehensive solution set. This approach ensures we identify all possible solutions to the cubic equation, encompassing both real and complex numbers.