Question

If a +b = 14 and ab = 25, find a4+b4​

Answer 1

Required Answer:-

Given:

• a + b = 14
• ab = 25

To find:

• The value of a⁴ + b⁴

Solution:

Given that,

➡ a + b = 14

Squaring both sides, we get,

➡ (a + b)² = 14²

➡ a² + 2ab + b² = 196

➡ a² + b² + 2 × 25 = 196

➡ a² + b² + 50 = 196

➡ a² + b² = 196 - 50

➡ a² + b² = 146

Again, squaring both sides, we get,

➡ (a² + b²)² = (146)²

➡ a⁴ + b⁴ + 2a²b² = 21316

➡ a⁴ + b⁴ + 2(ab)² = 21316

➡ a⁴ + b⁴ + 2 × 25² = 21316

➡ a⁴ + b⁴ + 2 × 625 = 21316

➡ a⁴ + b⁴ + 1250 = 21316

➡ a⁴ + b⁴ = 21316 - 1250

➡ a⁴ + b⁴ = 20,066

Hence,

➡ a⁴ + b⁴ = 20,066

Answer:

• a⁴ + b⁴= 20,066

Identity Used:

➡ (a + b)² = a² + 2ab + b²

More Identities:

➡ (a - b)² = a² - 2ab + b²

➡ a² - b² = (a + b)(a - b)

➡ (a + b)² = (a - b)² + 4ab

➡ (a - b)² = (a + b)² - 4ab

➡ 4ab = (a + b)² - (a - b)²

➡ (a + b)³ = a³ + b³ + 3ab(a + b)

➡ (a - b)³ = a³ - b³ - 3ab(a - b)