2. Reduce a, b, c, and d, to a common index. Ans. (a3)†, (b2)*, (4)†, and (14)†.

3. Reduce va + x, Va-x, and Va2-22, to a common index.

12

Ans. (a + x),

4. Reduce V, V2, and

(ax), and (a2 — x2)3.

5/3, to a common index.

Ans. 2V, VI, and 5√27.

279

5. Reduce ax, (bx), (cx), and (dx)1, to a common

index.

Ans. (a2x2)TM¥, (box€)15, (c424)15, and (d31⁄23)TE.

6. Reduce ca2, (das), and (24), to a common index. Ans. (cx2)‡, (dæ3)‡, and (24)‡.

7. Reduce 7, 10, and 11, to a common index. Ans. 49, 1000, and 101.

8. Reduce √, V, and V1331, to a common index. Ans. √ √, and √11.

II, FUNDAMENTAL OPERATIONS ON RADICALS.

1o. Addition of Radicals.

130. Radicals cannot be added unless they are similar. To determine when they are similar, we must reduce them to their simplest form; ical parts are the same, they will be

then, if their rad

similar, and if we

regard the common radical part as a unit, we shall have the following rule for finding their sum:

RULE.

Reduce the radicals to their simplest forms; then, if they are similar, add the coefficients for a new coefficient, and write the sum before the common radical part.

EXAMPLES.

Find the sums of the following groups of radicals:

1. √18, √32, √50, and √72.

Ans. 3√2 +4√/2 + 5√2 + 6√2 = 18√/2.

5. x√12a1x, 2a2√/27x3, 3a√/48a2x3, and 5a2x√3x.

Ans. 25a2x3x.

6. 54a+3, a16a-366, and /2a+ 4n+9

Ans. (3a2b+262 +an+3) /2a".

7. 6√4a2, 2V/2a, and Sa3.

Ans. 9/2a.

8. 2√3, 1√12, 4√/27, and 2√ 16. Ans. 13.

9. 36/2α3b2, 7/2ab5, and 8a2a2b3.

Ans. 18ab2a2b2.

131. We cannot subtract one radical from another unless the two are similar. In that case, we have the following

RULE.

Reduce the radicals to their simplest forms; then, if they are similar, subtract the coefficient of the subtrahend from that of the minuend, and write the remainder before the common radical part.

3. From Va3+2ab+ab2, subtract Va3-2a2b+ab2.

Ans. 2b√a.

4. From VV, subtract & V.

Ans. V6.

5. From √289ab, subtract √/144a3b.

Ans. 5a√b.

6. From 2/8a3+5√/72a3, subtract a√/18a+√/50ab2.

132. Since two radicals can always be reduced to a common index, we may take ab, and cd, to represent any two radicals whatever. The indicated product is,

We may change the order of the factors without changing the value of the product; hence, may write the product under the form,

n

But, from principle 1°, W × V = bd; hence,

a b × c vā

X = acŵbd;

whence the following

RULE.

Reduce the radicals to a common index; then multiply the coefficients together for a new coefficient, and the quantities under the radical signs for a new quantity under the radical sign, leaving the index unchanged.

By combining the above rule with that for the multiplication of polynomials, complicated radical expressions may be multiplied together.

1

3

1

1

3

1

48

48

96