To solve √((_a_ × _b_)/(_c_)) = _x_. Set _c_ on B to _a_ on A, and under _b_ on B read _x_ on D. To solve (_a_ × _b_)/(√_̅c_) = _x_. Set _c_ on B to _b_ an D, and under _a_ on C read _x_ on D. To solve √((_a_^2 × _b_)/(_c_)) = _x_. Set _c_ on B to _a_ on D, and under _b_ on B read _x_ on D. To solve (_a_^2 × _b_^2)/(_c_) = _x_. Set _c_ on B to _a_ on D, and over _b_ on C read _x_ on A. To solve (_a_√_̅b_)/(_c_) = _x_. Set _c_ on C to _b_ on A, and under _a_ on C read _x_ on D. To solve ((_a_ × √_̅b_)/(_c_))^2 = _x_. Set _c_ on C to _a_ on D, and over _b_ on B read _x_ on A. HINTS ON EVALUATING EXPRESSIONS. As a general rule, the use of cubes and higher powers should be avoided whenever possible. Thus, in the foregoing section, we recommend treating an expression of the form _a_√(_b_^3) as _a_ × _b_ × √_̅b_; the magnitudes of the values thus met with are more easily appreciated by the beginner, and mistakes in estimating the large numbers involved in cubing are avoided. EX.—7·3 × √(57^3) = 3140. Set 1 on C to 57 on D; bring cursor to 57 on B (R.H., since 57 has an _even_ number of digits); bring 1 on C to cursor, and under 7·3 on C read 3140 on D. As a rough estimate we have √(57), about 8; 8 × 57, about 400; 400 × 7, gives 2800, showing the result consists of 4 figures. An expression of the form _a_∛(_b_^2), or _a_ _b_^⅔, is better dealt with by rearranging as _a_ × (_b_)/(∛_b_). EX.—3·64∛(4·32^2) = 9·65. Set cursor to 4·32 on A, and move the slide until 1·63 is found simultaneously under the cursor on B and on D under 1 on C; bring cursor to 1 on C; 4·32 on C to cursor, and _over_ 3·64 on D read 9·65 on C. (Note that in this case it is convenient to read the answer on the _slide_; see page 22). From the slide rule we know ∛(4·32) = about 1·6; this into 4·32 is roughly 3; 3·64 × 3 is about 10, showing the answer to be 9·65. Similarly products of the form _a_ × _b_^{⁴⁄₃} are best dealt with as _a_ × _b_ × ∛_b_. Factorising expressions sometimes simplifies matters, as, for instance, in _x_^4 − _y_^4 = (_x_^2 + _y_^2)(_x_^2 − _y_^2). Here, working with the fourth powers involves large numbers and the troublesome determination of the number of digits in each factor; but squares are read on the rule at once, the number of digits is obvious, and, in general, the method should give a more accurate result. Take the expression, D_{1} = ∛((D^4 − _d_^4)/(D)) giving the diameter D_{1} of a solid shaft equal in torsional strength to a hollow shaft whose external and internal diameters are D and _d_ respectively. Rearranging as D_{1} = ∛(((D^2 + _d_^2)(D^2 − _d_^2))/(D)) and taking, as an example, D = 15 in. and _d_ = 7 in., we have D^2 + _d_^2 = 274 and D^2 − _d_^2 = 176; hence D_1 = ∛((274 × 176)/(15)) = ∛(3210) = 14·75 in.