Final answer:
To simplify the expression 4k⁴ / 8k⁶ + 8k⁹ / 4k⁹, we divide each term separately and find a common denominator before combining them. The equivalent expression is 5 / 2.
Step-by-step explanation:
To simplify the expression 4k⁴ / 8k⁶ + 8k⁹ / 4k⁹, let's first simplify each term separately. In the first term, 4k⁴ / 8k⁶, we can simplify both the numerator and denominator by dividing both terms by 4k². This gives us 1 / 2k². In the second term, 8k⁹ / 4k⁹, we can simplify by dividing both terms by 4k⁹, which gives us 2. Now, we can add these simplified terms together to get 1 / 2k² + 2. Since the two terms have different denominators, we need to find a common denominator. The common denominator is 2k², so we multiply the first term by k² / k² and the second term by 2k² / 2k² to get (1 * k²) / (2k²) + (2 * 2k²) / (2k²). This simplifies to k² / 2k² + 4k² / 2k², which further simplifies to (k² + 4k²) / 2k². Finally, we can combine like terms in the numerator to get (5k²) / (2k²). The k² terms cancel out, leaving us with 5 / 2. Therefore, the expression 4k⁴ / 8k⁶ + 8k⁹ / 4k⁹ is equivalent to 5 / 2.