Question

A+b =5 ab=-3 (a^4+b^4)×(a^3+b^3)×(a^2+b^2)

Answer 1

Нам нужно выразить каждую из скобок через сумму и произведение.

Делаем это:

$a^4+b^4=a^4+2(ab)^2+b^4-2(ab)^2=(a^2+b^2)^2-2(ab )^2=\\ =(a^2+2ab+b^2-2ab)^2-2(ab)^2=((a+b)^2-2ab)^2-2(ab)^2=\\ =(5^2-2\cdot(-3))^2-2(-3)^2=31^2-18=961-18=943$

$a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)(a^2+2ab+b^2-2ab-ab)=\\ =(a+b)((a+b)^2-3ab)=5\cdot(5^2-3\cdot(-3))=5\cdot34=170$

$a^2+b^2=a^2+2ab+b^2-2ab=(a+b)^2-2ab=5^2-2\cdot(-3)=31$

Осталось лишь перемножить эти три числа:

$943\cdot170\cdot31=4969610$

Answer 2

To find the value of the expression (a^4+b^4)×(a^3+b^3)×(a^2+b^2), we need to use the given information about the sum and product of a and b.

Given: a + b = 5 ab = -3

Let's first find the values of a^2 + b^2, a^3 + b^3, and a^4 + b^4, and then proceed with the multiplication:

1. a^2 + b^2: We can use the identity (a + b)^2 = a^2 + 2ab + b^2 and substitute the known values: (5)^2 = a^2 + 2(-3) + b^2 25 = a^2 - 6 + b^2 a^2 + b^2 = 25 + 6 a^2 + b^2 = 31

2. a^3 + b^3: We can use the identity (a + b)(a^2 - ab + b^2) = a^3 + b^3 and substitute the known values: (5)(a^2 - (-3) + b^2) = a^3 + b^3 5(31 + 3) = a^3 + b^3 5 * 34 = a^3 + b^3 a^3 + b^3 = 170

3. a^4 + b^4: We can use the identity (a^2 + b^2)(a^2 - ab + b^2) = a^4 + b^4 and substitute the known values: (31)(31 + 3) = a^4 + b^4 31 * 34 = a^4 + b^4 a^4 + b^4 = 1054

Now, let's find the value of the expression (a^4 + b^4) × (a^3 + b^3) × (a^2 + b^2):

(1054) × (170) × (31) = 56,334,740

So, the value of the expression (a^4+b^4) × (a^3+b^3) × (a^2+b^2) is 56,334,740.