Question

if a+b=3,a2+b2=7 then a4+b4=

Answer 1

# a+b = 3

a^2 + b^2 + 2ab = 3^2

2ab = 9 - 7

2ab = 2

ab = 1

a^2b^2 = 1

# a^2 + b^2 = 7

a^4 + b^4 + 2a^2b^2 =7^2

a^4 + b^4 = 49 - 2

a^4+b^4 = 47

Answer 2

Concept: An equation is said to be an identity if it is true regardless of the value of each of the variables. The equations that have the left side of the equation equaling the right side of the equation exactly for all possible values of the variable are known as algebraic identities.

The following four common algebraic identities are listed:

1st Identity: Square of the Sum of Two Terms in Algebra

(a+b)²=a²+2ab+b²

2nd Identity: Square of the Difference of Two Terms in Algebra

(a-b)²=a²-2ab+b²

3rd Identity: Algebraic Identity of Square Difference

(a+b)(a–b)=a²–b²

4th Identity: Identity in algebra (x+a)(x+b)

(x+a)(x+b)=x²+(a+b)x+ab

Given: a+b = 3, a²+b²=7

Find: a⁴+b⁴

Solution: We have that

a+b = 3, a²+b²=7

⇒ $a^2 + b^2 + 2ab = 3^2$

⇒2ab = 9 - 7

⇒2ab = 2

⇒ab = 1

⇒a²b² = 1

Now,

(a²+b²)² = a⁴ + b⁴ + 2a²b² =7²

⇒a⁴ + b⁴ = 49 - 2

⇒a⁴+b⁴ = 47

Hence, a⁴+b⁴=47

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