Can You Solve This Easy Algebra Math Problem?
Algebra Math Problem: The sum of two variables and the sum of the square of three variables are given. Then find the sum of the fourth power
Solve the following algebra math problem
a+b=1
a^2+b^2=9
a^4+b^4=?
Solution: we can solve this algebra math problem in two different ways
Method 1: Using the expansion of (a + b)^2
We know, a + b = 1
So,
(a + b)² = 1 = a² + 2ab + b²
ie,
1 = 9 + 2ab
So
ab = –4
Then we get
(a² + b²)² = a⁴ + 2a²b² + b⁴
81 = a⁴ + b⁴ + 2(ab)² = a⁴ + b⁴ + 2 × 16
So
a⁴ + b⁴ = 49
Method 2: Using the expansion of (a + b)^4
We know
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ = a⁴ + b⁴ + 4ab(a² + b²) + 6a²b²
1⁴ = a⁴ + b⁴ + 4(–4) × 9 + 6( –4)²
So
a⁴ + b⁴ = 49