Question

What is the simplified form of 1 - 25y² / 1+10y⁻¹ + 25y⁻² ?

a) 1 - 5y⁻¹ / 1 + 5y⁻¹
b) 1 - 5y⁻¹ / 1 - 5y⁻¹
c) 1 + 5y⁻¹ / 1 - 5y⁻¹
d) 1 + 5y⁻¹ / 1 + 5y⁻¹

Answer 1

The expression is simplified by recognizing patterns that match the difference of squares and a perfect square, resulting in the answer of option (b) 1 - 5y⁻¹ / 1 - 5y⁻¹.

Step-by-step explanation:

The question involves simplifying a complex rational expression. To simplify the expression 1 - 25y² / (1+10y⁻¹ + 25y⁻²), we can notice a pattern similar to the difference of squares. By recognizing the terms as part of (1 - 5y)(1 + 5y) = 1 - 25y² and (1 + 5y⁻¹)(1 + 5y⁻¹) = 1+10y⁻¹ + 25y⁻², we can match the numerator with the difference of squares and the denominator as a perfect square.

Thus, the simplified form is:

• (1 - 5y) / (1 + 5y⁻¹), which can be written as
• 1 - 5y⁻¹ / (1 + 5y⁻¹) after distributing the negative sign in the numerator.

This corresponds to option (b) 1 - 5y⁻¹ / 1 - 5y⁻¹ as the correct answer.