Problem in compound proportion, page 294:

EXPLANATION: The first statement shows the number of days it takes 12 men to do a piece of work; the second shows the number of days it takes 1 man to do the same work—that is, to load of the cargo; the third shows the time it would take 1 man to load

of the cargo; the fourth shows the number of days it would take him to load the whole cargo, or 1 ship; the fifth shows the number of days it would take him to load 3 ships; and the sixth shows the number of days it would take 5 men to load 3 ships. This is what was required, and the value of the fraction is 13 days.

3 days×12×5×3=131 days.

4X2X5

268. Rules for Analysis.

In order to make the analysis perfect and the writing simple, observe the following:

1. Always write, first, the number of the same kind with the answer sought. All the other numbers enter as multipliers or divisors.

2. Take but one step at a time.

3. Enter no mixed numbers or decimal fractions.

4. As soon as the statement is complete, reduce the fraction to its simplest form.

The observation of these rules is illustrated in the solutions given.

It would be well for the student to solve the other problems in proportion by this method of analysis.

INVOLUTION.

269. Any number multiplied by itself is said to be squared, or raised to the second power.

What is the second power of 5? Of 9? Of 11? When a number is used as a factor three times, the product is the third power of that number, or its cube. What is the cube of 3? Of 5? Of 7? Of 10?

So when a number is used as a factor four times, the product is the fourth power of that number, and

so on.

How many times must a number be used as a factor to obtain its fifth power? Its eighth power? Its tenth power?

Find the third power of 5. Of 7. Find the sixth power of 2. Of 5. Of 6.

Raising numbers to their powers is called Involution. A little figure showing how many times the number is to be used as a factor is called an Exponent. It is written at the right of, and a little above, the figure it affects.

34-81. In this example, what is the exponent? How many times must 3 be used as a factor? How many multiplications are necessary to find the fourth power of a number? The fifth power? The eighth

power?

Find the powers of the numbers indicated below:

73? 210? 35-? 56? 44-? 98=? 106=?

EVOLUTION.

270. When the power is given and we are asked to find one of its equal factors, the process is Evolution, and the number which is found is a Root.

One of the two equal factors that make a number is called the square root of that number. One of its three equal factors is called its cube root; one of four is its fourth root, and so on.

The radical sign, made thus, V, before a number, signifies that a root of that number is to be taken. A little figure written within the sign, called an index, shows what root is to be taken; thus, signifies the cube root. Usually the index for the square root is not written, but is understood; thus, V25 signifies the square root of 25, or 5. 81 ? √32=? 256? V125=?

As we find the areas of squares by multiplying two equal factors, and the contents of cubes by multiplying three equal factors, text-books on arithmetic properly give rules for extracting square roots and cube roots. Higher roots are extracted by using what are called logarithms; such a process belongs to higher mathematics. One who had much of such work to do would also use logarithms in extracting square and cube roots. We attempt no explanation of the rules for extracting square and cube roots; for such an explanation, we must resort to algebra. Explanations by diagrams and blocks explain only applications of the process, not the general principles.

SQUARE ROOT.

271. Rule for Square Root.

1. Begin at the decimal point and strike off two figures each time until you can recognize the greatest square in what remains.

2. Subtract that square from what was not pointed off, and write its root like a quotient in long division.

3. Bring down the next period to the remainder, for a new dividend.

4. Double the root already found and use it as a trial divisor, omitting the right-hand figure of the new dividend.

5.

Write the quotient in the root, and write the same figure at the right-hand of the divisor.

6. Multiply the complete divisor by the number just expressed in the root, and subtract the product thus obtained from the dividend.

7. To the remainder bring down the next period, and proceed as before.

It must be noticed that the squares of numbers as far as 9 must be thoroughly committed to memory, or we shall not be able to recognize the greatest square in the left-hand period. If the squares are known beyond the square of 9, we may be able to stop pointing off at an earlier stage, as we do in Example 2, next page.

Decimals must be pointed off toward the right, beginning at the decimal point; and all the periods must be full; if necessary, annex a zero to fill the last period. The number of decimal places in the root will equal the number of periods. Mixed decimals must be pointed both ways from the decimal point.

The square root of common fractions will be found by extracting the square root of numerator and of denominator.

The use of the trial divisor will sometimes give too large a figure for the root. When this is found to be the case, a smaller figure must be taken.

If the root can not be exactly found, decimal periods of zeros may be added to any desired extent, and so an approximate root may be found.

EXAMPLE 1.

51'40'89 (717

49

ILLUSTRATION:

We recognize 49 as the greatest square in 51. Our new dividend is 240, and our trial divisor is 14. The quotient of 24 divided by 14 is 1. The complete divisor is 141. The new dividend is 9,989, and the new trial divisor is 142. The quotient of 998 divided by 142 is 7, etc. The square root is found to be 717.

In Example 2 we recognize 144 as the greatest square in 156; hence, it is not necessary to point off any farther.