An Introduction to Algebra

CASE IX.

To find the roots of surd quantities.

RULE.

When the surd is a simple quantity, multiply its index by for the square root, by for the cube root, &c. and it will give the root of the surd part; which being annexed to the root of the rational part, will give the whole root required. And if it be a compound quantity, find its root by the usual rule. (x)

EXAMPLES.

1. It is required to find the square root of 93/3. Here (92/3)*=9a×3a×±=9a×3*=34/3 Ams,

2. It is required to find the cube root of√2.

Here ({√2)3=({})3× (2±×3)=1(2*)=1&/2 Ans. _

3. It is required to find the square root of 103.

4. It is required to find the cube root of

2791.

5. It is required to find the 4th root of a

(x) The nth root of the mth power of any number a, or the mtl

power of the nth root of a, is an

Also, the nth root of the mth root of any number a, or the mtl

root of the nth root of a, is a

From which last expression, it appears that, that the square root of the square root of a is the 4th root of a; and that the cube root of the square root of a, or the square root of the cube root of a, is the 6th root of a; and so on for the fourth, fifth, or any other numerical root of this kind.

6. It is required to find the cube root of

a α

7. It is required to find the square root of x-4x/α +4a.

8. It is required to find the square root of a+2/ab +b.

CASE X.

To transform a binomial, or a residual surd, into a

general surd.

RULE.

Involve the given binomial, or residual, to a power corresponding with that denoted by the surd; then set the radical sign of the same root over it, and it will be the general surd required.

EXAMPLES.

1. It is required to reduce 2+✔✔3 to a general surd. Here (2+3)=4+3+4√/3=7+43; therefore 2+√3=√7+4/3, the answer.

2. It is required to reduce 2+3 to a general surd. Here (√2+√3)2=2+3+2√√/6 = 5+2✅/6 ;

therefore √2+3=√5+2√6, the answer.

3. It is required to reduce 3/2+3/4 to a general surd.

Here (3/2+/4)3=6+63/2+63/4; therefore 3/2 +3/4 3/6(1+2+4, the answer.

4. It is required to reduce 3-5 to a general surd. 5. It is required to reduce √2-26 to a general surd.

6. It is required to reduce 4-7 to a general surd. 7. It is required to reduce 23/3-33/9 to a general. surd.

CASE XI.

To extract the square root of a binomial, or residual surd.

RULE.

Substitute the numbers, or parts, of which the given surd is composed, in the place of the letters, in one of the two following formulæ, according as it is a binomial or a residual, and it will give the root required.

Where it is to be observed, that if both a and a2-b, in these formulæ, be rational quantities, the root will consist either of two surds, or of a rational part and a surd, which are the only cases of the rule that are useful.

EXAMPLES.

1. It is required to find the square root of 11+√72, or √11+6√2.

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Whence

11+6√2=3+√2, the answer required.

2. It is required to find the square root of 3—2√2.

-1;

Whence 3-22√2-1, the answer required.
3. It is required to find the square root of 6±2√5.
Ans. √5±1

4. It is required to find the square root of 23±87.
Ans. 4±7.

5. It is required to find the square root of 36±10 ✓11.

6. It is required to find the square root of 33±12/6.

CASE XII.

To find such a multiplier, or multipliers, as will make any binomial surd rational.

RULE.

1. When one or both of the terms are any even roots, multiply the given binomial, or residual, by the same expression, with the sign of one of its terms changed; and repeat the operation in the same way, as long as there are surds, when the last result will be rational.

2. When the terms of the binomial surd are odd roots, the rule becomes more complicated; but for the sum or difference of two cube roots, which is one of the most useful cases, the multiplier will be a trinomial surd, consisting of the squares of the two given terms and their product, with its sign changed.

EXAMPLES.

1. To find a multiplier that shall render 5+3 rational.

Given surd 5+√3

Multiplier 5-1/3

Product 25-3=22, as required.

2. To find a multiplier that shall make 5+ √3 rational.