Part 17

1. 2[4^(1/3)] by 3[6^(1/3)].

2. 2^(1/2) by 3^(1/3).

3. 2^(1/4) by 4^(1/6).

4. [a + x^(1/2)]^(1/2) by [a - x^(1/2)]^(1/2).

5. 2^(1/2) + 3^(1/2) - 5^(1/2) by 2^(1/2) - 3^(1/2) + 5^(1/2).

6. -p/2 + ([p^2 - 4q]^(1/2))/2 by -p/2 - ([p^2 - 4q]^(1/2))/2.

Divide:

7. 27^(1/2) by 3^(1/2).

8. 4[18^(1/2)] by 5[32^(1/2)].

9. 3^(1/3) by 6^(1/2).

10. 3^(1/2) by 3^(1/4).

11. 6[105^(1/2)] + 18[40^(1/2)] - 45[12^(1/2)] by 3[15^(1/2)]. (Short division.)

12. 10^(1/3) - 4^(1/3) + 5^(1/3) by 3^(1/3).

Rationalize the denominator:

13. 2/(3^(1/2)); 7/(7^(1/2)); 5/(2[5^(1/2)]); 3/([a^2]^(1/5)); 4/([a^3]^(1/7)).

14. 2/(2^(1/2)) + 3^(1/2)); (a^(1/2) + b^(1/2))/(a^(1/2) - b^(1/2)); 3/(3 - 3^(1/2)).

15. [3^(1/2) + 2^(1/2)]/[6^(1/2) + 3^(1/2) - 2^(1/2)].

Review the method of finding the square root of a binomial surd. (By inspection preferably.) Then find square root of:

16. 5 + 2[6^(1/2)].

17. 17 - 12[2^(1/2)].

18. 7 - 33^(1/2).

~Reference:~ The chapter on Radicals in any algebra, beginning at Addition and Subtraction of Radicals.

MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS

Results by inspection, examples 1-10.

Divide:

1. (x^(5/17) + y^(5/17))/(x^(1/17) + y^(1/17)).

2. (x - y)/(x^(1/3) - y^(1/3)).

3. (m^2 + n^2)/(m^(2/3) + n^(2/3)).