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); [54]^(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 [11]^(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)? [23]^(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 [ab]^(1/n) = a^(1/n) . b^(1/n). Or, a^(1/n) . b^(1/n) = [ab]^(1/n). From this also ([ab]^(1/n))/(a^(1/n)) = b^(1/n). Multiply: 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[12]^(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[18]^(1/3) - 4[60]^(1/3) + 5[100]^(1/3) by 3[30]^(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)).