Certainly! Let's solve the equation step by step:
[tex]\[ x^2 - \sqrt[a]{xy} + y^2 = a^2 \][/tex]
Step 1: Understanding the equation
- [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are variables.
- [tex]\( \sqrt[a]{xy} \)[/tex] represents the [tex]\( a \)[/tex]-th root of the product [tex]\( xy \)[/tex].
- [tex]\( a \)[/tex] is a given constant.
Step 2: Rearrange the equation
We'll rewrite the equation in a more recognizable form to better understand its components. The given equation is:
[tex]\[ x^2 - (xy)^{1/a} + y^2 = a^2 \][/tex]
Step 3: Analyze individual terms
- [tex]\( x^2 \)[/tex] is the square of [tex]\( x \)[/tex].
- [tex]\( y^2 \)[/tex] is the square of [tex]\( y \)[/tex].
- [tex]\( (xy)^{1/a} \)[/tex] is the [tex]\( a\)[/tex]-th root of the product [tex]\( xy \)[/tex].
- [tex]\( a^2 \)[/tex] is the square of the constant [tex]\( a \)[/tex].
Step 4: Combine terms
Put all the terms together:
[tex]\[ x^2 + y^2 - (xy)^{1/a} = a^2 \][/tex]
This is the simplified and rewritten form of the given equation.
As a result, we obtained:
[tex]\[ x^2 + y^2 - (xy)^{1/a} = a^2 \][/tex]
This shows how the equation can be understood and rewritten. This simplified form might help in further analyzing the behavior of the equation or in solving for specific values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( a \)[/tex].
Note: Depending on the problem context, further steps might involve solving for one of the variables given the others, analyzing the behavior of the equation graphically, or using it in a wider application.
