Part 12

To solve √((a)/(b)) = x. Set b on B to a on A, and under 1 on C read x on D.

To solve a (b)/(c^2) = x. Set b on C to c on D and over a on B read x on A.

To solve c√((a)/(b)) = x. Set b on B to a on A, and under c on C read x on D.

To solve (√̅a)/(b) = x. Set b on C to a on A, and under 1 on C read x on D.

To solve (a)/(√̅b) = x. Set b on B to a on D, and under 1 on C read x on D.

To solve b√̅a = x. Set 1 on C to b on D, and under a on B read x on D.

To solve √(a^3) = x. Treat as a√̅a.

To solve a√(b^3) = x. Treat as a√̅b × b.

To solve (√̅a^3)/(b) = x. Treat as (√̅a × a)/(b).

To solve √((a^3)/(b)) = x. Treat as (√̅a × a)/(√̅b) = √((a)/(b)) × a.

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 D1 = ∛((274 × 176)/(15)) = ∛(3210) = 14·75 in.

Reversed Scale Notation.—With expressions of the form 1 − x, or 100 − x, it is often convenient to regard the scales as having their notation reversed, i.e., to read the scale backwards. When this is done the D scale is read as shown on the lower line—

Direct Notation 1 2 3 4 5 6 7 8 9 10 D Scale Reversed Notation 9 8 7 6 5 4 3 2 1 0

The new reading can be found by subtracting the ordinary reading from 1, 10, 100, etc., according to the value assigned to the R.H. index, but actually it is unnecessary to make this calculation, as with a little practice it is quite an easy matter to read both the main and subdivisions in the reversed order. Applications are found in plotting curves, trigonometrical formulæ, etc.