Understanding the Algebraic Problem

The question presents an algebraic expression involving the division of terms with cube roots and variables \(x\) and \(y\). We are told that the result of this division is a quadratic expression in \(x\) and \(y\) of the form \(Ax^2 + By^2 + Cxy\). Our task is to find the numerical values of the coefficients A, B, and C, and then use these values to calculate the value of the expression \(2A - 3B - 2\sqrt{6}C\).

Simplifying the Expression using Algebraic Identity

The expression to be simplified is:

\((16\sqrt{2}x^3 + 81\sqrt{3}y^3) \div (2\sqrt{2}x + 3\sqrt{3}y)\)

Let's look at the structure of the numerator and the denominator. The expression resembles the form of the sum of cubes identity, which is \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). If we divide \(a^3 + b^3\) by \((a+b)\), the result is \(a^2 - ab + b^2\).

Let's try to identify 'a' and 'b' in our given expression. Consider the denominator \((2\sqrt{2}x + 3\sqrt{3}y)\). Let:

• \(a = 2\sqrt{2}x\)
• \(b = 3\sqrt{3}y\)

Now, let's check if the numerator \((16\sqrt{2}x^3 + 81\sqrt{3}y^3)\) is indeed \(a^3 + b^3\) with these values of 'a' and 'b'.

• Calculate \(a^3\): \(a^3 = (2\sqrt{2}x)^3 = 2^3 \times (\sqrt{2})^3 \times x^3 = 8 \times (2\sqrt{2}) \times x^3 = 16\sqrt{2}x^3\). This matches the first term of the numerator.
• Calculate \(b^3\): \(b^3 = (3\sqrt{3}y)^3 = 3^3 \times (\sqrt{3})^3 \times y^3 = 27 \times (3\sqrt{3}) \times y^3 = 81\sqrt{3}y^3\). This matches the second term of the numerator.

Since the expression is in the form \((a^3 + b^3) \div (a+b)\), the result of the division is \(a^2 - ab + b^2\).

Calculating the Resulting Quadratic Expression

Now we substitute the expressions for 'a' and 'b' back into the formula \(a^2 - ab + b^2\):

• Calculate \(a^2\): \(a^2 = (2\sqrt{2}x)^2 = (2\sqrt{2})^2 x^2 = (2^2 \times (\sqrt{2})^2) x^2 = (4 \times 2) x^2 = 8x^2\).
• Calculate \(ab\): \(ab = (2\sqrt{2}x)(3\sqrt{3}y) = (2 \times 3) \times (\sqrt{2} \times \sqrt{3}) \times xy = 6\sqrt{6}xy\).
• Calculate \(b^2\): \(b^2 = (3\sqrt{3}y)^2 = (3\sqrt{3})^2 y^2 = (3^2 \times (\sqrt{3})^2) y^2 = (9 \times 3) y^2 = 27y^2\).

So, the result of the division is \(8x^2 - 6\sqrt{6}xy + 27y^2\).

Determining the Coefficients A, B, and C

We are given that the result of the division is \(Ax^2 + By^2 + Cxy\). By comparing the expression we found with the given form, we can identify the coefficients A, B, and C:

\(Ax^2 + By^2 + Cxy = 8x^2 + 27y^2 - 6\sqrt{6}xy\)

Matching the coefficients of the corresponding terms:

• The coefficient of \(x^2\) is A on the left side and 8 on the right side. So, \(A = 8\).
• The coefficient of \(y^2\) is B on the left side and 27 on the right side. So, \(B = 27\).
• The coefficient of \(xy\) is C on the left side and \(-6\sqrt{6}\) on the right side. So, \(C = -6\sqrt{6}\).

Calculating the Value of \(2A - 3B - 2\sqrt{6}C\)

Now we need to find the value of the expression \(2A - 3B - 2\sqrt{6}C\) using the values of A, B, and C we found:

\(A = 8\), \(B = 27\), \(C = -6\sqrt{6}\)

Substitute these values into the expression:

\(2A - 3B - 2\sqrt{6}C = 2(8) - 3(27) - 2\sqrt{6}(-6\sqrt{6})\)

Perform the calculations:

• \(2(8) = 16\)
• \(3(27) = 81\)
• \(-2\sqrt{6}(-6\sqrt{6}) = (-2) \times (-6) \times (\sqrt{6} \times \sqrt{6}) = 12 \times 6 = 72\)

Substitute these results back into the expression:

\(16 - 81 + 72\)

Finally, calculate the sum:

\(16 - 81 + 72 = (16 + 72) - 81 = 88 - 81 = 7\)

The value of \(2A - 3B - 2\sqrt{6}C\) is 7.

Revision Table: Summary of Findings

Additional Information: Working with Square Roots in Algebra

When working with square roots in algebraic expressions, remember these properties:

• \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\) (for non-negative a, b)
• \((\sqrt{a})^2 = a\) (for non-negative a)

In this problem, we used \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\) and \((\sqrt{6})^2 = 6\). Also, remember that the cube of a term like \(c\sqrt{d}\) is \((c\sqrt{d})^3 = c^3 \times (\sqrt{d})^3 = c^3 \times d\sqrt{d}\). For example, \((2\sqrt{2})^3 = 2^3 \times (\sqrt{2})^3 = 8 \times 2\sqrt{2} = 16\sqrt{2}\), and \((3\sqrt{3})^3 = 3^3 \times (\sqrt{3})^3 = 27 \times 3\sqrt{3} = 81\sqrt{3}\). These rules are essential for simplifying expressions involving radicals.