Finding HCF from LCM and Expressions

We are given two algebraic expressions and their Least Common Multiple (LCM). We need to find their Highest Common Factor (HCF).

Let the two expressions be:

• $A = 20x^3y^2$
• $B = 10x^4y^4$

The given LCM is:

• $\text{LCM}(A, B) = 20x^4y^4$

Using the HCF-LCM Relationship

There is a fundamental relationship between two numbers (or algebraic expressions) and their HCF and LCM:

$ A \times B = \text{HCF}(A, B) \times \text{LCM}(A, B) $

We can rearrange this formula to solve for the HCF:

$ \text{HCF}(A, B) = \frac{A \times B}{\text{LCM}(A, B)} $

Calculation Steps

1. Substitute the given values into the formula:

   $ \text{HCF} = \frac{(20x^3y^2) \times (10x^4y^4)}{20x^4y^4} $

2. Simplify the expression:

   First, simplify the coefficients:

   $ \frac{20 \times 10}{20} = 10 $

   Next, simplify the variable $x$ using exponent rules ($\frac{x^m \times x^n}{x^p} = x^{m+n-p}$):

   $ \frac{x^3 \times x^4}{x^4} = x^{3+4-4} = x^3 $

   Finally, simplify the variable $y$ using exponent rules:

   $ \frac{y^2 \times y^4}{y^4} = y^{2+4-4} = y^2 $

3. Combine the simplified parts:

   The HCF is the product of the simplified coefficient and variables:

   $ \text{HCF} = 10x^3y^2 $

Conclusion

The HCF of $20x^3y^2$ and $10x^4y^4$ is $10x^3y^2$. This corresponds to Option D.