3. Required the sum of ✔27 and √48. 4. Required the sum of 50 and 72. 5. Required the sum of√3 and √75.

Ans. 73.

Ans. 112.

Ans. 4

or 15.

Ans. 53/7.

Ans. 3/2.

6. Required the sum of 3/56 and 3/189. 7. Required the sum of 3/4 and 3/3

8. Required the sum of 3

ab and 516a b.

PROBLEM V.

To find the Difference of Surd Quantities.

PREPARE the quantities the same way as in the last rule; then subtract the rational parts, and to the remainder annex the common surd, for the difference of the surds required.

But if the quantities have no common surd, they can only be subtracted by means of the sign

EXAMPLES.

1. To find the difference between 320 and 80.

First,

320√64 × 5=8/5; and/S016 x 545.

Then 8/5-45=4/5 the difference sought.

2. To find the difference between 3/128 and 3/54.

First, 3/128=3/64 x 2=43/2; and 3/54 3/27 × 2=33/2. Then 43/2-33/23/2, the difference required.

3. Required the difference of 75 and
4. Required the difference of 3/256 and 3/32.
5. Required the difference of √ and
6. Required the difference of 3/3 and 3/25.
7. Find the difference of 24a2b2 and √✓✓54ab4.

48.

Ans. 3.

Ans. 23/4.

Ans. 6.

Ans. 75.

Ans. (a-2b)(3b2 — 2ab)√✅/ 6a.

PROBLEM VI.

To Multiply Surd Quantities together.

REDUCE the surds to the same index, if necessary; next multiply the rational quantities together, and the surds together; then annex the one product to the other for the whole product required; which may be reduced to more simple terms if necessary.

EXAMPLES.

EXAMPLES.

1. Required to find the product of 4/12 and 3/2. Here, 4× 3 × 12 × √√/2=12/12x2=12/24=12/4x6 = 12 × 2 × √6 246, the product required.

=

2. Required to multiply by 33.

Here ×V×V} = h×V÷z=iz×3/4=× ×3/18 = 18, the product required.

3. Required the product of 3/2 and 2√/8.

4. Required the product of 3/4 and 33/12. 5. To find the product of

Ans. 24. Ans. 13/6.

and. Ans.

6. Required the product of 23/14 and 33/4.

7. Required the product of 243 and a3.

15.

Ans. 123/7.

Ans. 2a2.

8. Required the product of (a + b) and (a + b)3⁄4. 9. Required the product of 2x + √/b and 2x — √✅b. 10. Required the product of (a + 2√/b), and (a — 2√/b)3.

I

I

11. Required the product of 2x and 3x”.

I

12. Required the product of 4x

and 2y*.

PROBLEM VII.

To Divide one Surd Quantity by another.

REDUCE the surds to the same index, if necessary; then take the quotient of the rational quantities, and annex it to the quotient of the surds, and it will give the whole quotient required; which may be reduced to more simple terms if requisite.

EXAMPLES.

1. Required to divide 6/96 by 3/8.

Here 63. (96 ÷ 8) =2√✓/12 = 2√/ (4 × 3) = 2 × 2√3 = 4/3, the quotient required.

2. Required to divide 123/280 by 33/5.

Here 123 = 4, and 2805568 × 7 = 23.7;
Therefore 4 x 2 x783/7, is the quotient required.

To Involve or Raise Surd Quantities to any Power.

RAISE both the rational part and the surd part. Or multiply the index of the quantity by the index of the power to which it is to be raised, and to the result annex the power of the rational parts, which will give the power required.

EXAMPLES.

1. Required to find the square of a3.

9

First, (4)2=2×4=1%, and (a2)2=až× 2—a2=a.

I

Therefore (a) = a, is the square required.

2. Required to find the square of a3.

First, × = 1, and (a3)2 = a3 = a3⁄4⁄a;

2

Therefore (ža3‍)3 = a/a is the square required.

3. Required to find the cube of √6 or 3 × 63.

First, (3)3 = × × ÷ = 287, and (64)3 =
3 2o7, and (67)3 = 62 = 6/6 ;

Theref. (6)3 = 27 × 6√/6 = 6, the cube required.

4. Required the square of 23/2.

5. Required the cube of 32, or 3.

√3.

6. Required the 3d power of √3.

7. Required to find the 4th power of √√2.

Ans. 43/4.

Ans. 3/3.

"

8. Required to find the mth power of a".

9. Required to find the square of 2 + √3.

PROBLEM IX.

To Evolve or Extract the Roots of Surd Quantities*.

EXTRACT both the rational part and the surd part. Or divide the index of the given quantity by the index of the root to be extracted; then to the result annex the root of the rational part, which will give the root required.

EXAMPLES.

1. Required to find the square root of 166. First, √16 = 4, and (6)2= 6÷2 = 64;

I

I

theref. (166) = 4.644/6, is the sq. root required.

2. Required to find the cube root of√3.

I

First, 3/2, and (√3)3 = 3÷÷3 = 37;

27 =

I

theref. (2√3)=.3/3, is the cube root required.

3. Required the square root of 63.

Ans. 6/6.

4. Required the cube root of a3b.

Ans. a3/6.

5. Required the 4th root of 16a2.

Ans. 2a.

I

6. Required to find the mth root of .x.

7. Required the square root of a2 - 6a√b + 9b.

The square root of a binomial or residual surd, a + b, or -b, may be found thus: Take a2

Thus, the square root of 4+ 2√3 = 1 + √3 ;

and the square root of 6-25 = √5 −1.,

But for the cube, or any higher root, no general rule is known.

INFINITE

2

INFINITE SERIES.

AN Infinite Series is formed either from division, dividing by a compound divisor, or by extracting the root of a compound surd quantity; and is such as, being continued, would run on infinitely, in the manner of a continued decimal fraction.

But, by obtaining a few of the first terms, the law of the progression will be manifest; so that the series may thence be continued, without actually performing the whole operation.

PROBLEM L.

To Reduce Fractional Quantities into Infinite Series by Division.

DIVIDE the numerator by the denominator, as in common division; then the operation, continued as far as may be thought necessary, will give the infinite series required.