Page:Elementary algebra (1896).djvu/205

187 Rh Find the sixth roots of the following expressions :

6. 1 +6x+ 15x^ + 20x^3 + 15x4 + 6x5 + x6. 7. x6 - 12 ax5 + 240 a4x2 - 192 a^5x + 60 a^2x^4 - 160 a^3x^3 + 64 a^6. 8. a6 - 18 a^5x + 135 a4x2 - 540 a^3x^3 + 1215 a2x4 - 1458 ax^5 + 729 x^6.

Find the eighth roots of the following expressions :

9. x8 - 8 x^7y + 28 x6y2 - 66 x^5y^3 + 70 x4y4 - 56 x^3y^5 +28 x^2y^6 - 8 xy^7 + y^8. 10. {x4 + 2(p - 1)x3 + (p2 -2p - 1)x2 - 2(p - 1)x + 1}4.

Find to four terms the square root of 11. a^2 - x, 12. x^2 + a^2. 13. a^4 - 3a^2. 14. 9 a^2 + 12 ax.

Find to three terms the cube root of

15. 1-6x + 21x^2. 16.27x^6-27x^5-18x^4. 17. 64-48x + 9x^2.

18. Find the fifth root of a10 - 10 a9 + 50 a8 - 160 a7 + S60 a^5 - 592 a5 + 720 a4 - 640 a^3 + 400 a2 - 160 a + 32.

19. a10 - 5 a9 + 20 a8 - 50 a7 + 105 a^6 - 161 a5 + 21O a4 - 200 a^3 + 160 a2 - 80 a + 32. 20. a10 + 5 a9 + 5 a8 - 10 a^7 - 15 a6 + 11 a^5 + 15 a4 - 10 a^3 — 5a2 + 5 a - 1.

205. Square and Cube Root of Numbers. Before leaving the subject of Evolution it may be useful to remark that the ordinary rules for extracting square and cube roots in Arithmetic are based upon the algebraic methods we have explained in the present chapter.

Ex. 1. Find the square root of 5329.

Since 5329 lies between 4900 and 6400, that is between (70)^2 and (80)^2, its square root consists of two figures and lies between 70 and 80. Hence, corresponding to a, tlie first term of the root in the algebraic process of Art. 198, we here have 70.

The analogy between the algebraic and arithmetical methods will be seen by comparing the cases we give below.

a^2 + 2 ab + b^2(a + b 5329(70 + 3 = 73. a^2 4900 2a + b 2ab + b^2 140 + 3 = 143 429 2 ab + b^2 429