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A radical sign, written thus V, and read square root, is used to express the square root of any number, before which it is placed. The same sign with the index of the root written over it, is used to express the other roots: thus V cube root: biquadrate root: ý fifth root: &c. We will give the following radical expressions; ✓9=3 V8=2; 8=3; $ 32=2; these expressions are read thus; the square root of 9 is equal to 3; the cube root of 8 is equal to 2; the fourth root of 81 is equal to 3; the fifth root of 32 is equal to 2. Hence it is evident that ✓9 Xv9=9; Ø 8 X 8 X 98=8; &c.

The explanation, which we have given of irrational numbers or surds, will readily enable us to apply to them the known methods of calculation. We know that the square root of 3 multiplied by itself must produce 3, which may be thus expressed, V3 XV3=3; also 3X3X3=3; V XV=*; V.5XV.5=.5;

XX=; V5X75X5X5=5; 1/2 X 12=2.

Instead of the radical sign, a fractional exponent is also used to express the roots of numbers. The numerator indicates the power of the number, and the denominator the root. Thus, 4. expresses the square root of 41, or 4; 4, the cube root of 4; 4, the biquadrate root of 4; 4, the fifth root of 4; 89, the cube root of the second power of 8; and since the second power of 8 is 64, and the cube root of 64 is 4, the expression 88 is equal to 4. The expression 4'=8, is read thus, the square root of the third power of 4 is equal to 8. The expression of is equivalent to V9: and gå is equivalent to 183: also 4* is equivalent to 42: the expression 4* is also equal to 41, because is equal to 1.

A line, or vinculum, drawn over several numbers, sig. nifies that the numbers under it are to be considered jointly! thus, V25+11 is equal to 6, because 25+11 is

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36, and the square root of 36 is 6; but 25 +11 is equal to 16, because the square root of 25 is 5, and 5+11 is 16. The expression » 27—6+43 is equivalent to V64. And $100—73=3., Also 20-V9+7+1=17. Likewise 90–944+53—45+6=13.

XXIX.

EXTRACTION OF THE SQUARE ROOT.

The product of a number multiplied by itself, is called a square; and for this reason, the number, considered in relation to such a product, is called a SQUARE ROOT. For example, when we multiply 12 by 12, the product, 144, is a square, and 1? is the square root of 144.

If the root contains a decimal, the square will also contain a decimal of double the number of places; for example, 2.25 is the square of 1.5; and vice versa, if the square contains a decimal, the square root will contain a decimal of half the number of places; for instance, 1.5 is the square root of 2.25.

In the upper line of the following table are arranged several square roots, and in the lower line, their squares. Square roots. / 112131415161718 9 | 10 | 11 | 12

Squares. |1|4|9|16|25|36 49 64 31/100 121/144 When the square of a mixed number is required, it may be reduced to an improper fraction, then squared, and reduced back to a mixed number,

The squares of the numbers from 3 to 5, increasing by ), are as follows. Square roots. 331 | 31 | 3 | 4 | 41 | 41 | 4}

Squares. 9 (10% 12 (141/16/187/20}|221 From this table we infer, that if a square contains a fraction, its square root also contains one; and vice versa,

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It is not possible to extract the square root of any number, which is not a perfect square; we can approximate the square root of such numbers, however, by the aid of decimals.

When a root is composed of two or more factors, we may multiply the squares of the several factors together, and the product will be the square of the whole root; and conversely, if a square be composed of two or more factors, each of which is a square, we have only to multiply together the roots of those factors, to obtain the complete root of the whole square. For example, 2304

4X 16X 36; the square roots of the factors are 2, 4, and 6; and 2 X 4X6=48; and 48 is the square root of 2304, because 48 X 48=2304.

A square number cannot have more places of figures than double the places of the root; and, at least, but one less than double the places of the root. Take, for instance, a number consisting of any number of places, that shall be the greatest possible, of those places, as 99, the square of which is 9801, double the places of the root. Again, take a number consisting of any number of places, but let it be the least possible, of those places, as 10, the square of which is 100, one less than double the places of the root.

As the places of figures in the square cannot be more than double the number of places in the root, whenever we would extract the square root of any number, we point it off into periods of two figures each, by placing a dot over the place of units, another over hundreds, &c. Thus 1936. The places in the root can never be more or .ess in number, than the number of periods thus pointed off. When the number of places in the given sum is an odd number, the left hand period will contain only one figure, as i69; but the root will nevertheless consist of as many places as there are periods; for 13 is the

square root of 169.

The terms, square and square root, are derived from geometry, which teaches us that the area of a square is found by multiplying one of its sides by itself.

The word AREA signifies the quantity of space contained in any geometrical figure.

A SQUARE is a figure having four equal sides, and all its angles right angles. If we suppose the length of a side of the annexed square to be four feet, it is evident that the figure contains 4 times 4 small squares, each of which is 1 foot in length and 1 foot in breadth; and since a foot in length and a foot in breadth constitute a square foot, the whole square contains 4 times 4, or 16 square feet. If, instead of 4 feet, the length of a side were 4 yards, the whole square would contain 16 square yards; &c. Hence it is evident that the area, which is 16, is found by multiplying a side of the square by itself.

A PARALLELOGRAM is an oblong figure, having two of its sides equal and parallel to each other, but not of the same length with the other two, which are also equal and parallel to each other.

We find the area, or contents of a parallelogram by multiplying the length by the breadth. If we suppose the annexed right angled parallelogram to be 8 feet long and 2 feet wide, it is manifest that it contains 2 times 8, or 16 square feet; if the length were 8 yards and the breadth 2 yards, it would contain 16 square yards; if 8 miles long and 2 miles wide, 16 square miles; &c. We see that the area of this parallelogram is the same with that of the preceding square; therefore the square root of the area of a parallelogram gives the side of a square-equal in area to the parallelogram.

It is further to be observed, that the square root of the area of any geometrical figure whatever, is the side of a square, equal in area to the figure.

When the area of a square is given, the process of finding one of its sides, which is the root, is called the extraction of the square root, the principles of which we will now proceed to explain.

We have already learned, that a square number is a product resulting from two equal factors. For example, 2025 is a square number resulting from the multiplication of 45 by 45. To investigate the constituent parts of this product, we will separate the root into two terms, thus, 40+5, and multiply it by itself in this form. We begin with multiplying 40+5 by 5, and set down the products separately, which are 200 +25; we then multiply 40+ 5 by 40, and set down the products separately, which are 1600 + 200; the whole product, therefore, is 1600+ 200 + 200 + 25 = 2025; thus we see, that the whole product or square contains the 'square of the first term, 40 X 40=1600; twice the product of the two terms, 40 X 5 X 2=400; and the square of the last term, 5 X 5=25.

Now the extraction of the square root is the reverse of squaring or raising to the second power; therefore, the operation of extracting the square root of 2025, which we know is the square of 45, must be performed in the inverted order of raising 45 to the second power.

We will now extract the square root of 2025, and explain the process, step by step.

2025 (45

16 Divisor. 40 X 2=80 425 dividend Divisor, increased by last fig. 85 425 product of 85 by 5.

Explanation of the process. We began by separating the given number into periods of two figures each, putting a dot over the place of units, and another over hundreds, and thereby ascertained that the root would contain two places of figures. We then found that the greatest square in the left hand period was 16, and placed its root, which is 4, in the quotient, and subtracted the square from the left hand period, and to the remainder brought down the next period for a dividend.

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