Question

Suppose that some nonzero real numbers a and b satisfy 1/a + 1/b = 1/2 and a + b = 10. Find the value of a³ + a³​

Answer 1

The value of a³ + a³​ is 400

Explanation:

1/a + 1/b = 1/2 , a + b = 10

so,

1/a + 1/b = 1/2

multiplying by ab on both sides,

(ab)/a + (ab)/b = ab/2

b + a = ab/2

2(a+b) = ab,

Since a+b = 10, we get,

ab = 2(10)

ab =20

Now, since we know ab, and a+b, we can find the value of a^2 + b^2 in the following way,

since we know that,

`(a+b)^2=a^2+2ab+b^2,\\so,\\(10)^2=a^2+b^2+2(20)\\100=a^2+b^2+40\\100-40=a^2+b^2\\60=a^2+b^2`

Hence 60=a^2+b^2

Now, finally, we put all this in the formula,

`a^3 + b^3 = (a + b)(a^2 + b^2 - ab)\\a^3+b^3 = (10)(60-20)\\a^3+b^3=(10)(40)\\a^3+b^3=400`

Hence the value of a³ + a³​ is 400