Page:Elementary algebra (1896).djvu/26

8 2. The sign $$=$$ should never be used except to connect quantities which are equal. Beginners should be particularly careful not to employ the sign of equality in any vague and inexact sense.

3. Unless the expressions are very short the signs of equality in the steps of the work should be placed one under the other.

4. It should be clearly brought out how each step follows from the one before it; for this purpose it will sometimes be advisable to add short verbal explanations; the importance of this will be seen later.

If $$a = 2$$, $$b = 3$$, $$c = 1$$, $$d = 0$$, find the numerical value of

1. $$6a + 5b - 8c + 9d$$.

2. $$3a - 4b + 6c + 5d$$.

3. $$6ab - 3d + 2a - 5cb + 2db$$.

4. $$abc + bcd + cda + dab$$.

5. $$3abc - 2bcd + 2cda - 4dab$$.

6. $$2bc + 3cd - 4da + 5ab$$.

7. $$3bc + 5cda - 7dab + abc$$.

8. $$a^2 + b^2 + c^2 + d^2$$.

9. $$2a^2 + 3b^3 - 4c^4$$.

10. $$a^4 + b^4 + c^4$$.

If $$a = 1$$, $$b = 2$$, $$c = 3$$, $$d = 0$$, find the numerical value of

11. $$a^3 + b^3 + c^3 + d^3$$.

13. $$3abc-b^2c - 6a^3$$.

14. $$2a^2 + 2b^2 + 2c^2 + 2d^2 - 2bc - 2cd - 2da - 2ab$$.

15. $$a^2 + 2b^2 + 2c^2 + d^2 + 2ab + 2bc + \frac{2}{7}cd$$.

16. $$2c^2 + 2a^2 _ 2b^2 - 4cb + 6abcd$$.

17. $$13a^2 + \tfrac{11}{9}c^4 + 20ab - 16ac - 16bc$$.

18. $$6ab - \tfrac{4}{3}ac^2 - 2a + \frac{1}{8}b^4 - 3d + \frac{4}{9}c^3$$.

19. $$125b^4c-9d^5+3abc^2d$$.

12. $$\tfrac{1}{2}bc^3 - a^3 - b^3 - \frac{3}{4}ab^3c$$.

If $$a = 8$$, $$b = 6$$, $$c = 1$$, $$x = 9$$, $$y = 4$$, find the numerical value of

20. $$\tfrac{5}{3}a - \tfrac{1}{9}b^3 + \tfrac{7}{8}y^2$$.

21. $$\tfrac{5}{27}ax - \tfrac{32}{y^2} - \tfrac{6a}{cxy}$$.

22. $$\tfrac{3a^2b}{cxy^2} - \tfrac{5y}{a}$$.

23. $$\sqrt[3]{\left(\tfrac{6cy^4}{x^2}\right)} + 2\sqrt{\left(\tfrac{3a^3}{4b^3}\right)}$$.

24. $$\sqrt[3]{(bxy)}-\tfrac{1}{8}b^2+\tfrac{8x^2}{b^2y}$$.

25. $$\tfrac{3}{4}ac - \sqrt{\left(\tfrac{b^2}{9y}\right)} - \sqrt[3]{\left(\tfrac{by}{x^2}\right)}$$.

26. $$\tfrac{5b^2y^3}{12a^2x} - \sqrt[3]{\left(\tfrac{ax^4}{b^2y^2}\right)} + \sqrt{\left(\tfrac{ab^3}{3x}\right)}$$.