148. 1. Prices of two kinds of bicycles are such that 7 of one kind and 12 of the other kind can be obtained for $ 640. Three more of the latter can be purchased for $180 than can be purchased of the former for $120. Find the price of each.

2. A person lends $2500 in two separate sums at the same rate of interest. At the end of one year the first sum with interest amounts to $997.50; at the end of two years the second sum with (simple) interest amounts to $1705. Find the two separate sums and the rate of interest.

3. Prove that if the sum of two real numbers is multiplied by the sum of their reciprocals, the product cannot be less than 4.

Let x and y be the two real numbers, p the product considered; solve for the unknown ratio 2.

y

4. A traveler starts from A toward B, another traveler starts at the same time from B toward A. In two hours they meet 20 miles from A. When the second traveler arrives at A, the first is still 135 miles from B. Find the distance between A and B.

5. If in ax2 + bx + c = 0, the coefficients are related to each other in such a way that a + b = 2 and a = 2 c, what must be the value of a, so that 8 will be a root of the given quadratic equation?

6. The difference between two numbers is 12, and the difference between their cubes is 7488. What are the two numbers?

7. A sum of money at simple interest for four years amounts to $2240. Had the rate of interest been 1% higher, the sum would have amounted to $80 less than this in two years. Find the capital and the rate.

8. The sum of the areas of two circles is 694.2936 sq. in.; the sum of their radii is 21. Find their radii.

9. A circular track is constructed so that the width of the track is of the inside diameter. The area of the track is 2500 sq. yd. What are the inside and outside lengths of the track?

10. Find two numbers such that their sum is equal to their product and also to the difference of their squares.

11. Find two numbers such that the sum of the numbers is equal to the sum of their squares and also to twice their product.

12. Find two numbers whose sum is 8 and whose product is 25.

13. Find three numbers such that the sum of the squares of the first two is equal to three times the square of the first minus the square of the second, and is also equal to the first minus the second plus twice the third, and also to the sum of the three numbers.

CHAPTER VII

EXPONENTS, RADICALS, IMAGINARIES

MEANINGS OF DIFFERENT KINDS OF EXPONENTS

149. The different kinds of exponents which have been studied thus far have been interpreted in the following manner, m and n being taken to be positive integers :

While a number a has in general n different nth roots, it is agreed that

1

only one of them shall be represented by a" or √ā, namely, the so-called principal root. This restriction was made in order to avoid unnecessary complication and confusion in the interpretation of expressions and equations involving radicals. Accordingly, √4 = +2, −√4 = − 2, √8 = + 2, -8=-2, -8=-2, V-8 +2, √16 +2, etc.

==

− =+ =

But other

The exponents considered above are all rational numbers. exponents have been brought to our attention. In the study of logarithms, mention was made of the fact that logarithms are often irrational numbers. Since logarithms are really exponents, it follows that exponents may be irrational. In more advanced books still another kind of exponent is considered, namely, the exponent that is an imaginary number.*

*For a fuller treatment of the theory of exponents, consult H. B. Fine, College Algebra, 1904, p. 376.

OPERATIONS WITH EXPONENTS

150. Operations involving exponents are subject to the following laws:

DIFFERENT KINDS OF NUMBERS

151. In the study of arithmetic and algebra several different kinds of numbers have come to our notice.

These may be

Arithmetic deals with real numbers, all positive, but some of them irrational.

The irrational numbers of arithmetic arise in the process of finding roots.

In algebra use is made of the numbers encountered in arithmetic, but convenience forces upon us the need of considering also negative numbers for the purpose of indicating relations of opposition, as temperatures above or below a certain fixed point, debts or assets, distances to the right or left, etc. In the solution of quadratic equations we meet a still different type. If we try to solve x2 + 1 = 0, we are confronted with the symbol √— 1.

This result greatly embarrassed mathematicians of the eighteenth and of earlier centuries; no satisfactory explanation of it could be made at first. But now the symbols are recognized as constituting a new type of number, the so-called imaginary number, which deserves a legitimate place in algebra and is of great service in certain advanced developments of algebra, that are useful in the study of polyphase electric currents and of other advanced topics in mathematical physics. Just as negative numbers have been found truly useful in elementary algebra, so imaginary numbers have been found useful in advanced algebra. The symbol V-1, or this multiplied by any real number b, such as b√-1, is called a pure imaginary number.

Expressions of the type a b√-1, which are the sum or difference of a number a and of a pure imaginary bv-1 (a and b being any real numbers, but b0), are called complex numbers. The term "complex" was introduced because the parts of the expression are partly real and partly imaginary.

SIMPLIFYING RADICALS

152. A radical is said to be in its simplest form:

(a) When the index of the root is as small as possible,

(b) When the expression under the radical sign, called the radicand, is integral,

(c) When the radicand contains no factor with a negative exponent, or raised to a power equal to or greater than the index of the root.

√25 is not in its simplest form, because the index 4 of the root is not as small as possible; we have √25 = 5a = 51 = √5.

✓ is not in its simplest form, because the radicand, }, is fractional. Vab is not in its simplest form, because the exponent 4 is greater than 3, the index of the root; we have Va+b=avab.

V3 ab2 is in its simplest form, since it fulfills all three conditions.