EXTRACTION OF THE SQUARE ROOT OF A QUANTITY OF THE FORM a±√b.

(288.) Binomials of this class require particular attention, because they frequently occur in the solution of equations of the fourth degree, such as are treated of in Art. 184. Thus the equation

Hence, in order to find the value of x, we must extract the square root of the binomial 7±4√3.

In order to show that the square root of such an expression may sometimes be extracted, take the binomial

and find its square.

2±√3,

(2±√3)2=4±4√3+3=7±4√3.

Therefore, the square root of 7±4√3 is 2±√3.

The square root of an expression of the form ab may, therefore, sometimes be extracted, and it is required to determine a general method for this purpose whenever it is prac ticable.

THEOREM I.

(289.) The sum or difference of two surds can not be equal to rational quantity.

For, if possible, let √a±√b=c, where c denotes a rationa! quantity, and a, b denote surd quantities.

By transposing b and squaring both sides, we obtain a '2c√b+b; whence, by transposition and division, we have

The second member of the equation contains only rational quantities, while b was supposed to be irrational; that is, we find an irrational quantity equal to a rational one, which is absurd. Hence the sum or difference of two surds can not be equal to a rational quantity.

THEOREM II.

In every equation of the form

x±√y=a±√b,

the rational parts on the opposite sides are equal to each other, and also the irrational parts.

or

For if x is not equal to a, let it be equal to a±z

that is, a rational quantity is equal to the difference of two surds, which, by the last Theorem, is impossible. Therefore, x=a, and, consequently, √y=√b.

If

then will

THEOREM III.

√a+ √b is equal to x+√y,

Va-vb be equal to x-√y.

For, by involution, a+b=x2+2x√y+y.

2

But, by the last Theorem,

and

Subtracting, we obtain

a=x2+y,

√b=2x√y.

a-√b=x2-2x√y+y.

Therefore, by evolution, √a- √b=x−√y.

(290.) To find an expression for the square root of a±√b.

where Ρ and q may be both radicals, or one rational and the other a radical, but p2 and q2 are required to be rational. Then, by the last Theorem,

√a-√b=p-q (2).

Multiplying these equations together, we obtain

√ a2—b=p2—q3 (3), a rational quantity.

Hence we see that, in order that √a+ √b may be expressed by the sum of two radicals, or one rational term and the other a radical, the expression a'-b must be a perfect square.

Let, then, a2-b be a perfect square, and put √a2—b=c; equation (3) will thus become

p'-q'=c.

Squaring equations (1) and (2), we obtain

(291.) Hence, to extract the square root of a binomial of the form ab, we have the following

RULE.

From the square of the rational part (a), take the square of the irrational part (b); extract the square root of the remainder and, calling that root c, the required root will be

Ex. 1. Required the square root of 4+2√3.

Here a=4, and √b=2√3; therefore, a2-b=c2=16—12=4; or c=2. Hence, by the above formula, the required root will

The square of 3+1 is 3+2√3+1=4+2√3.

Q

Ex. 2. Required the square root of 11+6√2.

Here a=11, and √b=6√2; therefore, b=36×2=72; and a'-b=49=c'. Hence c=7, and we find the square root of 11+62 is 9+ √2, or 3+√2. Ans.

Ex. 3. Required the square root of 11-2√30.

Ans. √6-5

Ex. 4. Required the square root of 2+√3.

(292.) This method is applicable even when the binomial contains imaginary quantities.

Ex. 5. Required the square root of 1+4 √−3.

Here a=1, and √b=4√−3; hence b⇒-48, and a3—b=49; therefore, c=7. The required square root is √4+√−3=2+ √-3. Ans.

-

Ex. 6. Required the square root of }+{√ −3.

Ans. 1+1√−3.

Ex. 7. Required the square root of 2 √1.

Here we put a=0; hence c=2, and the required root is

which may be easily verified.

Ex. 8. Required the value of the expression

√6+2√5−√6−2√5.

Ex. 9. Required the value of the expression

√1+3√−20+√√4−3 √ −20.

Ans. 6.

SECTION XVIII.

INFINITE SERIES.

(293.) An infinite series is an infinite number of terms, each of which is derived from the preceding term or terms according to some law.

or

As

1+++++, &c.,

1-計一, &c.

27 81

These are examples of geometrical progressions, in the first of which the ratio is, and in the second it is -1.

Infinite series may arise from the common operations of division, the extraction of roots, and other processes of calculation, as will be seen hereafter.

A converging series is one in which the sum of any number of its terms is finite, as in the examples just given.

A diverging series is one in which the sum of its terms is not finite; as,

1+2+3+4+5+6+7, &c.

An ascending series is one in which the exponents of the unknown quantity continually increase; as,

ax+bx2+cx2+dx*+ex3+, &c.

A descending series is one in which the exponents of the unknown quantity continually decrease; as,

ax¬1+bx¬2+cx ̄¬3+dx¬+ex¬+, &c.

PROBLEM I.

(294.) Any series being given, to find its several orders of

lifferences.