The Expression x2/3 - 2 in Terms of k

Understanding how to transform and express mathematical equations can be a valuable skill for students and professionals alike. In this article, we explore how to rewrite the expression x2/3 - 2 in the form 1/3 x - kxk, where k is a positive constant. This process involves algebraic manipulation and the application of factoring techniques.

Step-by-Step Transformation

Let's begin with the given expression:

Original Expression

#x7b;x sup 2 sub 3 - 2

Step 1: Combine Terms Over Common Denominator

First, we combine the terms over a common denominator:

#x7b;x sup 2 sub 3 - 2 x sup 2 / 3 - 6 / 3 (x sup 2 - 6) / 3

Step 2: Factor the Numerator

Next, we factor the numerator, using the difference of squares formula:

x sup 2 - 6 x sup 2 - (sqrt{6}) sup 2 (x - sqrt{6})(x sqrt{6})

Step 3: Substitute Back into the Expression

We now substitute this back into the original expression:

(x sup 2 - 6) / 3 (x - sqrt{6})(x sqrt{6}) / 3

Step 4: Express in the Desired Form

We can now express the fraction in the desired form:

(1/3)x - sqrt{6}x * sqrt{6}

From this, we can see that k sqrt{6}.

The value of k is boxed{sqrt{6}}.

Alternative Approach

Alternatively, we can start by equating the given forms:

Expression in Terms of k

(x sup 2 / 3 - 2 1/3 x - kx k)

Step 1: Take LCM of Denominator on LHS and Cancel 3 from Both Sides

Taking the least common multiple (LCM) and simplifying, we get:

x sup 2 - 6 kx * x - k^2

Step 2: Use Difference of Squares Formula

Using the difference of squares formula, we can compare both sides:

x sup 2 - 6 x root 6 * x - root 6 (since a - b sup 2 (a - b) (a b))

Thus, we find that:

Comparing both sides, k root 6

Conclusion

The value of k is determined by equating the expressions and simplifying. Regardless of the method used, the final result is consistent:

k root 6

Final Answer

The value of k is: boxed{sqrt{6}}