Final answer:
To find the value of m in the equation y¹²+81=(y⁶+m)(y⁶-m), we can use the difference of squares formula. After rearranging the equation, we find that m = ±√(-81). However, since the square root of a negative number is not defined in the real number system, there is no real solution for m.
Step-by-step explanation:
To find the value of m in the equation y¹²+81=(y⁶+m)(y⁶-m), we can use the difference of squares formula, which states that a² - b² = (a + b)(a - b). In this case, we can rewrite the equation as y¹² + 81 = (y⁶ + m)(y⁶ - m). Comparing it to the difference of squares formula, we can identify that a = y⁶ and b = m. So, we have a² - b² = (a + b)(a - b), which gives us y¹² + 81 = (y⁶ + m)(y⁶ - m). Expanding the right side of the equation, we get y¹² + 81 = y¹² - m². Now, we can cancel out the y¹² term from both sides and solve for m². Subtracting y¹² from both sides, we get 81 = -m². Finally, taking the square root of both sides, we find that m = ±√(-81). Since the square root of a negative number is not defined in the real number system, there is no real solution for m.