Part 16

9. Find the square root of: 25a^(4/3)b^(-3) - 10a^(2/3)b^(-3/2) - 49 + 10a^(-2/3)b^(3/2) + 25a^(-4/3)b^3.

10. Simplify [(2^(n + 2))/(4^(-n)) / (8^n)/(2^3)]^(1/5).

11. Find the value of (7 . 13^0 / 7)/(21^0) + 3^0 x (4^0 . 7^0)/[(7a + b)^0] + 8^(-2/3).

12. Express as a power of 2: 8^3; 4^5; 4^3 . 8^(2/3) . 16^(3/4).

13. Simplify {[(x^(a + 1))/(x^(1 - a))]^a / [(x^a)/(x^(1 - a))]^(a - 1)}^(1/(3a - 1)).

14. Simplify [(x^(5/2) y^(4/3))/(z^(-5/4)) . (z^4)/(x^(-3) y^(-5/3)) / (y^(-2) z^(1/4))/(x^(-1/2))]^(1/5).

15. Expand (a^(1/2) + b^(1/3))^4, writing the result with fractional exponents.

~Reference:~ The chapter on Theory of Exponents in any algebra.

RADICALS

1. Review all definitions in Radicals, also the methods of transforming and simplifying radicals. When is a radical in its simplest form?

2. Simplify (to simplest form): [2/3]^(1/2); [1/11]^(1/2); [3/5]^(1/3); 3[5/6]^(1/2); (2a/b)[(8b^2)/(27a)]^(1/2); [5/(x^n)]^(1/2n); (a + b)^2 [(-a^4)/((a + b)^5)]^(1/3); 27^(1/2); ^(1/3); -5[125^(1/2)].

3. Reduce to entire surds: 2[3^(1/2)]; 2[3^(1/4)]; 6[2^(1/3)]; a[[b^2]^(1/n)]; -3[2^(1/3)]; 3a[[(a + 2)/(6a^2)]^(1/3)]; (a + 2y)[(a - 2y)/(a + 2y)]^(1/2).

4. Reduce to radicals of lower order (or simplify indices): [a^2]^(1/4); [a^3]^(1/6); [27a^3]^(1/6); [81 a^4 x^8]^(1/12); [9x^2 y^4 z^10]^(1/2n).

5. Reduce to radicals of the same degree (order, or index): 7^(1/2) and ^(1/3); 5^(1/3) and 3^(1/4); 7^(1/6) and 3^(1/2); [x^m]^(1/n) and [x^n]^(1/m); [c^y]^(1/x), [c^z]^(1/y), and [c^x]^(1/z).

6. Which is greater, 3^(1/2) or 4^(1/3)? ^(1/3) or 2[2^(1/2)]?

7. Which is greatest, 3^(1/2), 5^(1/3), or 7^(1/4)? Give work and arrange in descending order of magnitude.

Collect:

8. 128^(1/2) - 2[50^(1/2)] + 72^(1/2) - 18^(1/2).

9. 2[5/3]^(1/2) + (1/6)60^(1/2) + 15^(1/2) + [3/5]^(1/2).

10. [(m - n)^2a]^(1/2) + [(m + n)^2a]^(1/2) - [am^2]^(1/2) + [a(n - m)^2]^(1/2) - a^(1/2).

11. A and B each shoot thirty arrows at a target. B makes twice as many hits as A, and A makes three times as many misses as B. Find the number of hits and misses of each. (Univ. of Cal.)

~Reference:~ The chapter on Radicals in any algebra (first part of the chapter).

The most important principle in Radicals is the following:

(ab)^(1/n) = a^(1/n) b^(1/n).

Hence ^(1/n) = a^(1/n) . b^(1/n).

Or, a^(1/n) . b^(1/n) = ^(1/n).

From this also (^(1/n))/(a^(1/n)) = b^(1/n).

Multiply: