Sum of the cube root of two numbers is 13 and difference of their cube roots is 1 find the numbers

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1 1
And 2 2

2 (* ,

=b, by the question, 16

16 Or.(a + x) + (a – X)* = 16b, by multiplication, Or 2a* + 12aRx2 + 2x4 = 166, by involution and addition, And 2.4 +62-22 = 8b - a*, by transposition and division, Whence 2* 3a+ (9a% + 8b – a)= - 3a* +

V 8 (a* + b), by the rule, And x = V [- 3a + 2 v 2 (at +6)], by extracting the root, Where, by substituting, 13 for a, and 4721 for b,

we shall have x = 3,

13 + x 16 Therefore

8, the greater number, 2

2 13

10 And

5, the less number. 2

2 The sum of which is 13, and 84 + 54 = 4721. 10. Given the sum of two numbers equal s, and their product p, to find the sum of their squares, cubes, biquadrates, &c.

and

denote the two numbers; then

(1.) x+y=s, (2.) xy =p. From the square of the first of these equations take twice the second, and we shall have

(3.) x2 + ya = 58 = sum of the squares. Multiply this by the 1st equation, and the product will be

23 + xya + x*y + y: = — 2 sp From which subtract the product of the first and second equations, and there will remain

(4.) x3 + y = 58 — 3sp = sum of the cubes. Multiply this likewise by the 1st, and the product will be 204 + xy + xoy + y4 == 54 — 3*p; from which subtract the product of the second and third equations, and there will remain

(5.) 2* + y = 5* - 4s*p + 2p = sum of the biquadrates. And, again multiplying this by the 1st equation and subtracting from the result the product of the second and fourth, we shall have