An exponential quantity is raised to any power by multiplying the exponent of the quantity by that of the power; and conversely, any root of an exponential quantity is extracted by dividing the exponent of the quantity by that of the root.

3m

For (a) = a.a = a2′′, (aTM)3 = aTM.aTM.aTM = a3TM, (aTM)1 = a1m, &c.

Or the nth power of am is aTM", and is found by multiplying m, the exponent of the quantity, by n, the exponent of the power.

In the same manner, (a”)TM = a””.

Hence (aTM)" = (a")TM. Conversely, (aTM")" = (a")TM" = a""= a”.

1

Or the nth root of an is a", and is found by dividing mn,

the ex

ponent of the quantity, by n, the exponent of the root. Also it may be shewn that (a")” — a ̄””, and (a ̄m)—" = (a−")−m. Any power of the product, and of the quotient of two quantities is equal to the product and quotient respectively of the powers of the

a

quantities; that is, (ab)" =a"b" and {%}'

=

a

bm

Also any root of the product, and of the quotient of two quantities is equal to the product and the quotient respectively of the roots of

(+1)=(-1)1. (−1) t = − ( − 1)t += − ( − 1)2 = − ( − 1) = + 1.
(+1)=+1. ( − 1); = + √/ − 1.

&c., &c. Hence they recur after the 4th power.

Generally, since every integral number is of the form 4m, 4m+1, 4m+2, 4m+3, · '. ( + √√ −1) 11 = [ { + ( − 1)}} ] = {( − 1)2 }" = ( + 1)" = + 1. (+√ −1)4m+1=(√−1)√-1=+1x√-1=+ √ −1. (+√-1)+2=(-1)(-1)+1x-1=-1.

(+√ −1)4m+3= (√ − 1) +m+2 √✓ − 1=-1×√-1=-√-1. Hence when n is of the forms 4m and 4m+2, ( + √ −1)"=±1,

and when n is of the forms 4m+1 and 4m+3, (+√ − 1)"=±√-1. In a similar way may be found the forms of the nth powers of √ -1. (−√ −1) = [{−(−1)} } + ]" = { +(−1)2 } TM = ( + 1)TM = + 1. (-1)+m+1=(-√-1) x (-√-1)=+1x-v-1=-V-1.

4m

(−√−1) 4m+2=(-√ − 1)+m. { − ( − 1)' } = +1. (−1) = +1x-1=-1.

(−√ −1)4m+3=( − √ − 1)4m+2 x ( -√-1)=-1x -√-1=+√ −1. Hence when n is of the form 4m and 4m+2, ( − √ − 1)” =±1; and when n is of the form 4m+1, and 4m+3, (−√ − 1)" = FV - 1.

6. 4x-4x3-3x2+2x+1. 7.x-10ax+33a2x2-40a3x+16a.

8. 49x+56xy+30x2y2+8xy3+y'.

9. a+2ab+3a*b2+4a3b3+3a2b*+2ab+bo.

10. a3+6a2+13a+14a3+12a* +10a3 +5a2+2a+1.

.

11. {(x-1)(x-2)+3} {(x-2)(x-3)+4}+8x+55x+175. 12. 16a (a+b+c)+4abc(b+c)+4a2(b2+c2)+16a2bc+b2c2.

13. 3(3a2-2ab+b2) (a2+3b2)+b2 (a+4b)2.

14. a2(a-5b)(a - b)+b2(3a-b2-3a2b2.

15. 4a2b2+(a2+b2)2+4ab(a2+b2).

16. a*+b*+c*+d1+2a2(b2+d2)+2b2 (c2+d2)+2c2(a2+d2). 17. (a—b)*+2(a'+b1)—2(a2+b2)(a2—b2).

18. 4a(2a3-a2-1)+(2a3 +a2+1).

19. ax(ax+1)(ax+2)(ax+3)+1.

20. a2(a+b2-c3)+2(a+b)(b+c)ac+2a2(ab+ac+bc)+b2c2. 21. (a2+b2+c2)3+2(ab+ac+be)3—6(a2+b2+c2)(ab+be+ac)2.

II.

Extract the square roots of the following numbers—

1. 625, 2704, 65536, 425104, 6589489, 58553104, 152399025. 2. 25, 0025, 000025, 2500, 250000, 10, 1, 01, C01, 0001, ⚫000001.

3. ·025, 520, 121, 1-2101, 4·9, 049, 490, 9, 101, ∙144, ·9, 54. 4. 17, 201, 121, 71%, 4987, 538182.

5. 3, 23, 11, 178, 15, 3.

6. 1-64, 1064, 1.0064, 1.00064, 1.000064, 1-0000064.

7. 6·4, 0·64, ·064, ·0064, ·00064, 000064, 64.00064, 64,000064.

8. 76212900, 00762129, 7.62129, 76212-9, 7621-29, 76-2129.

III.

1. Shew that every quantity has two equal square roots, one positive and the other negative. Find the square roots of a2—2ab+b2, and shew when a trinomial of this form is a complete square, that four times the product of the first and third terms is equal to the square of the second.

2. Show that (a+b+c)2=a2+b2+c2+2ab+2ac+2bc: and from this

result deduce the squares of a+b-c, a−b+c, −a+b+c, a-b-c respectively, without performing the operation of multiplication.

3. Write down the algebraical expression for the product of the square root of the sum of two quantities a and b, and the cube of their sum; and calculate its value to seven places of decimals when a = 1 and b = 2.

4. State in words the several operations to be performed in order to obtain the result expressed by the following algebraical expression, and find its value when a = b = 4.

5. If at arise from the products of +a× +a, and —a× —a; can a2 be considered as the only square root of a1?

9 21 49 6 16, and explain

14

why, if the terms be arranged in the reverse order, and the root be then extracted, a result will be obtained differing only in the sign of the whole quantity from that obtained in the first instance.

8. Find the square root of x-2xy+4y2 in terms of a and b, where 9a2+12ab, and y = 2b2+6ab.

9. Shew that the square root of (x−x ̄')2+(y—y-1) is a rational

1. What relations must subsist between a, b, c in order that ax2+bx+c may be a complete square?

2. Shew that if x =

m2-cn2 bn2-2amn'

then ax+bx+c is a perfect square.

3. After performing the operation for the extraction of the square root, find a value of x which will make x+6x+11x2+3x+31 a perfect square number.

4. Shew that '+2ax3+b2x2+c3x+d' is a perfect square if

x=

(a2 — b2)3 — 4 d1

4a (a2-b2)+c3°

5. Find the values of a, b, and e which will make the expression 26-8x+ax+bx3+cx2-44x+4 a perfect square.

6. If x+px+qx+rx+s be a perfect square, then shall r2 = p28, p+884q, and p3+8r=4pq.

7. ax2+2bxy+cy2+2dx+2ey+f is a perfect square if bac, d2 = af, 62=cf.

8. If ax1+bx3+cx be subtracted from (x2+2x+4)3, and the remainder be a perfect square; determine the values of a, b, and c.

V.

1. Find the number of which 1.4142... is the approximate square root.

2. Find the least number which when multiplied into 850 will make the result a perfect square.

3. Find the sum, difference, product, and quotient of 9+ √·4 and 9.4.

4. Shew that the integral part of the square root of 111111 is 333 with a remainder 222.

5. Shew that the square root of 3 differs from 12 by less than. 6. The difference between two square integral numbers is equal to the sum of their square roots, together with twice the sum of all the intermediate numbers between those square roots.

7. Shew whether the square root of any integer which is not a perfect square number can be expressed by means of an integer and a rational fraction.

8. Explain why no integer or terminating decimal can have a recurring decimal for its square root. In what case can a recurring decimal have such a root? Give an example.

9. In extracting the square root of any number, if more than half of the required number of digits be obtained by extraction, the remaining digits may be found by division: and determine the limit which the remainder cannot exceed after any step of the process.

VI.

1. Show that every square number ends in 1, 4, 5, 6, 9 or 00, and that the fourth powers of all numbers prime to 2 and 5 end in unity.

2. The square of 10129 is 102596641; find the square of 101293 without going through the operation of the multiplication.

3. Write an algebraical expression by means of which it may be shown how to find a series of square numbers, each of which shall be the sum of two square numbers.

4. If the difference of any two numbers be unity, the difference of their squares is equal to the sum of the two numbers.

5. The difference of the squares of a number composed of two digits and the number formed by reversing these digits, is equal to 99 times the difference of the squares of the digits.

6. Shew that the square of every odd number diminished by unity is divisible by 8, as also the difference of the squares of any two odd

numbers.

7. In any square number 4 is the only digit which can occupy both the place of units and of tens.

8. Show that any odd square number when divided by 4 will leave unity for the remainder. Is this property true of any other than square numbers ?

9. Prove that if any number which is a perfect square be divided by 3 it can never leave 2 for a remainder.

10. If the last digit but one in a perfect square be an odd number, the last digit must be 6.

11. Every square number is divisible by 5, or becomes so when increased or diminished by 1.

12. If a number end in 5, shew that the last two digits of the square of the number are 25, and the number composed of the remaining digits is equal to the product of two consecutive numbers.

13. A number consisting of four digits of which the two middle ones are zeros; prove that the difference of the squares of the number and of the number formed by reversing the digits is equal to 999999 times the difference of the squares of the extreme digits.

14. If two numbers are equidistant from 25, the square of the greater exceeds the square of the less by as many hundreds as the number itself exceeds 25; also if two numbers be equidistant from 50, the square of the greater exceeds that of the less by twice as many hundreds as the number itself exceeds 50. Shew how to apply these properties in readily finding the squares of all numbers from 13 to 100.

VII.

two square numbers

1. Shew that the product of the sum of any by the sum of two other square numbers can always be expressed by the sum of two square numbers (Diophantus).

2. If the product of any four consecutive natural numbers be increased by unity, and the product of any four consecutive odd numbers be increased by 16, the sums are square numbers.

3. If any three consecutive whole numbers be taken, prove that the sum of the squares of the greatest and least is greater by 2 than twice the square of the middle number.

4. If there be three numbers, one of which is the sum of the other two; twice the sum of their fourth powers is a square number.

5. Of any six consecutive numbers, the first being odd, the difference between the sum of the squares of the even and odd numbers is equal to three times the sum of the third and fourth numbers.

6. Shew that the product of six consecutive numbers cannot be a complete square.

=

7. If a, b, c be integers and a2+b22, then abc is always divisible by 60.