a^2+b^2=25,a^3+b^3=91.

Question

Find the value of a and b.

Answer 1

a² + b² = 5² reminds us about Pythagorean theorem (triangle rectangle) with a = 3 and b = 4 working !

 

Substituting ... 3³ + 4³ = 27 + 64 = 91 as desired. 

 

a³ + b³ = (a + b)(a² - ab + b²) = 7 (25 - 12) = 7 x 13 = 91

 

a = 3

b = 4

 

 

Answer 2

a^2+b^2=25

look at squares: 1, 4, 9, 16, 25,...

9 and 16 add up t0 25

a=3 or 4 and b=the other number

 

a^3+b^3=91

look at cubes: 1, 8, 27, 64, 125,...

27 and 64 add up to 91

a=3 or 4 and b=the other number

Answer 3

There is another real solution pair as well.  It is approximately 4.61313222 and -1.92849123

 

The final solution pair is complex.  -4.84232 +- 3.308791535 i