Ex. 4. Find the square root of 361a2b'x3.

Ans. 19ab'x*.

Ex. 5. Find the square root of 529a'm'x'y'.

Ans. 23am'x3y*.

Ex. 6. Find the square root of 841a*b*c*d3.

Ans. 29a'b'c'd.

(145.) It appears from the preceding rule that a monomial can not be a perfect square unless its coefficient be a square number, and the exponent of each of its letters an even number.

Thus 7ab' is not a perfect square, for 7 is not a square number, and the exponent of a is not an even number. Its square root may be indicated by the usual sign thus:

√7ab2.

Expressions of this kind are called surds, or radicals of the second degree.

A radical quantity is the indicated root of an imperfect power.

(146.) We have seen, Art. 121, that whatever may be the sign of a monomial, its square must be affected with the sign +. Hence we conclude that

If a monomial be positive, its square root may be either positive or negative.

Thus

√9a1=+3a2, or -3a2,

for either of these quantities, when multiplied by itself, produces 9a. We therefore always affect the square root of a positive quantity with the double sign , which is read plus or minus.

QUEST.-What is necessary in order that a monomial may be a per fect square? What are surd quantities? What sign must we prefix to the square root of a monomial ?

(147.) If a monomial be negative, the extraction of its square root is impossible, since we have just seen that the square of every quantity, whether positive or - negative, is necessarily positive.

are algebraic symbols indicating operations which it is impossible to execute. Quantities of this nature are called imaginary or impossible quantities, and are frequently met with in resolving equations of the second degree.

PROBLEM IV.

To reduce Radicals to their most simple Forms. (148.) Surds may frequently be simplified by the application of the following principle: the square root of the product of two or more factors is equal to the product of the square roots of those factors: Or, in algebraic language,

√ ab=√ax√b;

for each member of this equation, squared, will give the same quantity, viz., ab.

Let it be required to reduce √4a to its most simple form.

This expression may be put under the form √4

xva.

But 4 is equal to 2.

QUEST.-Can we extract the square root of a negative quantity? What are imaginary quantities? Upon what principle may radical quantities be simplified?

Hence

√4a=√4x√a=2√a.

2✓a is considered a simpler form than √4a. Again: reduce ✓48 to its most simple form.

✓48 is equal to √16×3=√16× √3=4√3. (149.) Hence, in order to simplify a monomial radical of the second degree, we have the following

RULE.

Separate the expression into two factors, one of which is a perfect square; extract its root; and prefix it to the other factor, with the radical sign between them.

In the expressions 2√ɑ and 4√3, the quantities 2 and 4 are called the coefficients of the radicals.

To determine whether a given number contains a factor which is a perfect square, try whether it is divisible by either of the square numbers 4, 9, 16, 25, 36, etc.

Examples.

1. Reduce √18a' to its most simple form.

2. Reduce √75a'b' to its most simple form.

Ans. 3a√2.

Ans. 5ab2 √3a.

Ans. 8a √3b.

3. Reduce ✓192ab to its most simple form.

4. Reduce 486a b'c' to its most simple form.

Ans. 9a'b'c √6a.

QUEST.-Give the rule for simplifying radical quantities. What are coefficients of radicals? How may we know whether a number con tains a factor which is a square?

5. Reduce √432a b' to its most simple form.

Ans. 12ab √36. 6. Reduce √1125a bc to its most simple form. Ans. 15a2c √ 5ab. 7. Reduce √343a'm'x to its most simple form. Ans. 7am 7ax. 8. Reduce √980a b'c to its most simple form. Ans. 14a b'c √5b. 9. Reduce √2560a'c'x to its most simple form. Ans. 16a'c v10ax. 10. Reduce ✓1331a'b'x to its most simple form. Ans. 11a'b v11abx.

11. Reduce 1864a'b'c' to its most simple form. Ans. 12ab'c' √6bc.

PROBLEM V.

To add Radical Quantities together.

(150.) Two radicals of the second degree are similar, when the quantities under the radical sign are the same in both.

Thus 3a and 5a are similar radicals.

So also are 2√3 and 5/3.

But 2/3 and 3√2 are not similar radicals. Radicals may be added together by the following

RULE.

When the radicals are similar, add their coefficients, and to the sum annex the common radical.

QUEST.—What are similar radicals? Give the rule for adding radical quantities.

But if the radicals are not similar, and can not be made similar by the reductions in the preceding Article, they can only be connected together by the sign of addition.

Ex. 1. Find the sum of 2a and 3a.

As these are similar radicals, we may unite their coefficients by the usual rule; for it is evident that twice the square root of a, and three times the square root of a, make five times the square root of a.

Ex. 2. Find the sum of 36 and 56.

Ans. 86.

Ex. 3. Find the sum of 2m ✅a and 3na.

Ans. (2m+3n)√a.

Ex. 4. Find the sum of 7 √3a and 9√3a.

Ans. 16 √3a.

Ex. 5. Find the sum of m√a+b and n√ a+b.

Ans. (m+n) va+b.

(151.) If the radicals are originally dissimilar, they must, if possible, be made similar by the method of Art. 149.

Ex. 6. Find the sum of 27 and 48.

Ex. 8. Find the sum of √75 and 108.

QUEST.-May radicals which appear to be dissimilar be sometimes nited?

Ans. 113.