To multiply quantities having the same base, add exponents.
To divide quantities having the same base, subtract exponents.
To raise a quantity to a power, multiply exponents.
To extract a root, divide the exponent of the power by the index of the root.
1. Find the value of
2. Find the value of
Give the value of each of the following:
3.
4. Express as some power of 7 divided by itself.
Simplify:
5. (Change to the same base first.)
6.
7.
8.
9.
10.
11.
Reference: The chapter on Theory of Exponents in any algebra.
1.
2.
Factor:
3.
4.
5.
6.
7. Find the H. C. F. and L. C. M. of
8. Simplify the product of:
and (Princeton.)
9. Find the square root of:
10. Simplify
11. Find the value of
12. Express as a power of 2:
13. Simplify
14. Simplify
15. Expand writing the result with fractional exponents.
Reference: The chapter on Theory of Exponents in any algebra.
RADICALS
1. Review all definitions in Radicals, also the methods of transforming and simplifying radicals. When is a radical in its simplest form?
2. Simplify (to simplest form):
3. Reduce to entire surds:
4. Reduce to radicals of lower order (or simplify indices):
5. Reduce to radicals of the same degree (order, or index): and and and and and
6. Which is greater, or ? or ?
7. Which is greatest, or ? Give work and arrange in descending order of magnitude.
Collect:
8.
9.
10.
11. A and B each shoot thirty arrows at a target. B makes twice as many hits as A, and A makes three times as many misses as B. Find the number of hits and misses of each. (Univ. of Cal.)
Reference: The chapter on Radicals in any algebra (first part of the chapter).
The most important principle in Radicals is the following:
Hence Or,
From this also
Multiply:
1. by
2. by
3. by
4. by
5. by
6. by
Divide:
7. by
8. by
9. by
10. by
11. by (Short division.)
12. by
Rationalize the denominator:
13.
14.
15.
Review the method of finding the square root of a binomial surd. (By inspection preferably.) Then find square root of:
16.
17.
18.
Reference: The chapter on Radicals in any algebra, beginning at Addition and Subtraction of Radicals.
MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS
Results by inspection, examples 1-10.
Divide:
1.
2.
3.
4.
Multiply:
5.
6.
7.
8.
9.
10.
Factor:
11.
12.
13.
14.
Factor, using radicals instead of exponents:
15.
16.
17. (factor as difference of two squares).
18. (factor as difference of two cubes).
19. (factor as difference of two fourth powers).
20. Find the H. C. F. and L. C. M. of
21. Solve (short method)
22. Simplify (Princeton.)
1. Solve for p:
2. Solve for t:
3. Find the square root of 8114.4064. What, then, is the square root of .0081144064? of 811440.64? From any of the above can you determine the square root of .081144064?
4. The H. C. F. of two expressions is and their L. C. M. is If one expression is what is the other?
5. Solve (short method):
6. Solve
7. Simplify
8. Does ? Does ?
9. Write the fraction with rational denominator, and find its value correct to two decimal places.
10. Simplify (Princeton.)
1. Rationalize the denominator of (Univ. of Cal.)
2. Simplify (Univ. of Penn.)
3. Find the value of when (Cornell.)
4. Find the value of x if
(M. I. T.)
5. A fisherman told a yarn about a fish he had caught. If the fish were half as long as he said it was, it would be 10 inches more than twice as long as it is. If it were 4 inches longer than it is, and he had further exaggerated its length by adding 4 inches, it would be as long as he now said it was. How long is the fish, and how long did he first say it was? (M. I. T.)
6. The force P necessary to lift a weight W by means of a certain machine is given by the formula
6. where a and b are constants depending on the amount of friction in the machine. If a force of 7 pounds will raise a weight of 20 pounds, and a force of 13 pounds will raise a weight of 50 pounds, what force is necessary to raise a weight of 40 pounds? (First determine the constants a and b.) (Harvard.)
7. Reduce to the simplest form:
8. Determine the H. C. F. and L. C. M. of and (College Entrance Board.)
1. Simplify
2. Simplify, writing the result with rational denominator:
(M. I. T.)
3. Find
4. Expand
5. Expand and simplify
6. Solve the simultaneous equations
(Yale.)
7. Find to three places of decimals the value of
when and (Columbia.)
8. Show that is the negative of the reciprocal of (Columbia.)
9. Solve and check
10. Assuming that when an apple falls from a tree the distance (S meters) through which it falls in any time (t seconds) is given by the formula (where ), find to two decimal places the time taken by an apple in falling 15 meters. (College Entrance Board.)
Excellent practice may be obtained by solving the ordinary formulas used in arithmetic, geometry, and physics orally, for each letter in turn.
Arithmetic
Geometry
Physics
QUADRATIC EQUATIONS
1. Define a quadratic equation; a pure quadratic; an affected (or complete) quadratic; an equation in the quadratic form.
2. Solve the pure quadratic
Review the first (or usual) method of completing the square. Solve by it the following:
3.
4.
5.
6.
Review the solution by factoring. Solve by it the following:
7.
8.
9.
10.
Solve, by factoring, these equations, which are not quadratics:
11.
12.
13.
Review the solution by formula. Solve by it the following:
14.
15.
16.
17.
Solve graphically:
18.
19.
Reference: The chapter on Quadratic Equations in any algebra (first part of the chapter).
1. Solve by three methods—formula, factoring, and completing the square:
Review equations in the quadratic form and solve:
2.
3.
4. (Let and substitute.)
5.
6.
Solve and check:
7.
8.
9.
Give results by inspection:
10.
11.
12. How many gallons each of cream containing 33% butter fat and milk containing 6% butter fat must be mixed to produce 10 gallons of cream containing 25% butter fat?
13. I have $6 in dimes, quarters, and half-dollars, there being 33 coins in all. The number of dimes and quarters together is ten times the number of half-dollars. How many coins of each kind are there? (College Entrance Board.)
Reference: The last part of the chapter on Quadratic Equations in any algebra.
THE THEORY OF QUADRATIC EQUATIONS
I. To find the sum and the product of the roots.
The general quadratic equation is
(1)
Or, (2)
To derive the formula, we have by transposing
Completing the square,
Extracting square root,
Transposing,
Hence,
These two values of x we call roots. For convenience represent them by and
| Hence, | |
| Adding, | (3) |
| Also, | |
| Multiplying, | (4) |
Hence we have shown that
and
Or, referring to equation (2) above, we have the following rule:
When the coefficient of is unity, the sum of the roots is the coefficient of x with the sign changed; the product of the roots is the independent term.
Examples:
1.
Sum of the roots
Products of the roots
2.
Sum of the roots
Product of the roots
3.
Sum of the roots
Product of the roots
II. To find the nature or character of the roots.
As before,
The determines the nature or character of the roots; hence it is called the discriminant.
If is positive, the roots are real, unequal, and either rational or irrational.
If is negative, the roots are imaginary and unequal.
If is zero, the roots are real, equal, and rational.
Examples:
1.
The roots are real, unequal, and irrational.
2.
The roots are imaginary and unequal.
3.
The roots are real, equal, and rational.
III. To form the quadratic equation when the roots are given.
Suppose the roots are 3, -7.
| Then, | Or, | ||
| Multiplying to get a quadratic, | |||
Or,
Or, use the sum and product idea developed on the preceding page. The coefficient of must be unity.
Add the roots and change the sign to get the coefficient of x.
Multiply the roots to get the independent term.
The equation is
In the same way, if the roots are the equation is
Find the sum, the product, and the nature or character of the roots of the following:
1.
2.
3.
4.
5.
6.
7.
8.
Form the equations whose roots are:
9. 5, -3.
10.
11.
12. -3, -5.
13.
14.
15.
16. Solve Check by substituting the values of x; then check by finding the sum and the product of the roots. Compare the amount of labor required in each case.
17. Solve
18. Is a perfect square?
19. Find the square root (short method):
20. Solve
21. The glass of a mirror is 18 inches by 12 inches, and it has a frame of uniform width whose area is equal to that of the glass. Find the width of the frame.
OUTLINE OF SIMULTANEOUS QUADRATICS
Simultaneous Quadratics
Case I.
One equation linear.
The other quadratic.
Method: Solve for x as in terms of y, or vice versa, in the linear and substitute in the quadratic.
Case II.
Both equations homogeneous and of the second degree.
Method: Let and substitute in both equations.
Alternate Method: Solve for x in terms of y in one equation and substitute in the other.
Case III.
Any two of the quantities given.
Method: Solve for and then add to get x, subtract to get y.
Case IV.
Both equations symmetrical or symmetrical except for sign. Usually one equation of high degree, the other of the first degree.
Method: Let and and substitute in both equations.
Special Devices
I. Consider some compound quantity like etc., as the unknown, at first. Solve for the compound unknown, and combine the resulting equation with the simpler original equation.
II. Divide the equations member by member. Then solve by Case I, II, or III.
III. Eliminate the quadratic terms. Then solve by Case I, II, or III.
SIMULTANEOUS QUADRATICS
Solve:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13. (Yale.)
14. (Princeton.)
15. Plot the graph of each equation. (Cornell.)
16. (Columbia.)
In grouping the answers, be sure to associate each value of x with the corresponding value of y.
17. The course of a yacht is 30 miles in length and is in the shape of a right triangle one arm of which is 2 miles longer than the other. What is the distance along each side?
Reference: The chapter on Simultaneous Quadratics in any algebra.
RATIO AND PROPORTION
1. Define ratio, proportion, mean proportional, third proportional, fourth proportional.
2. Find a mean proportional between 4 and 16; 18 and 50; and
3. Find a third proportional to 4 and 7; 5 and 10; and
4. Find a fourth proportional to 2, 5, and 4; 35, 20, and 14.
5. Write out the proofs for the following, stating the theorem in full in each case:
(a) The product of the extremes equals etc.
(b) If the product of two numbers equals the product of two other numbers, either pair etc.
(c) Alternation.
(d) Inversion.
(e) Composition.
(f) Division.
(g) Composition and division.
(h) In a series of equal ratios, the sum of the antecedents is to the sum of the consequents etc.
(i) Like powers or like roots of the terms of a proportion etc.
6. If write all the possible proportions that can be derived from it. [See (5) above.]
7. Given write the eight proportions that may be derived from it, and quote your authority.
8. (a) What theorem allows you to change any proportion into an equation?
(b) What theorem allows you to change any equation into a proportion?
9. If what is the ratio of x to g? of y to r? of y to g?
10. Find two numbers such that their sum, difference, and the sum of their squares are in the ratio 5 : 3 : 51. (Yale.)
Reference: The chapter on Ratio and Proportion in any algebra.
An easy and powerful method of proving four expressions in proportion is illustrated by the following example:
Given prove that
Let
Also
Substitute the value of a in the first ratio, and c in the second:
Then
Also
Axiom 1.
Or,
If prove:
1.
2.
3.
4.
5.
6. The second of three numbers is a mean proportional between the other two. The third number exceeds the sum of the other two by 20; and the sum of the first and third exceeds three times the second by 4. Find the numbers.
7. Three numbers are proportional to 5, 7, and 9; and their sum is 14. Find the numbers. (College Entrance Board.)
8. A triangular field has the sides 15, 18, and 27 rods, respectively. Find the dimensions of a similar field having 4 times the area.
ARITHMETICAL PROGRESSION
1. Define an arithmetical progression.
Learn to derive the three formulas in arithmetical progression:
2. Find the sum of the first 50 odd numbers.
3. In the series 2, 5, 8, ···, which term is 92?
4. How many terms must be taken from the series 3, 5, 7, ···, to make a total of 255?
5. Insert 5 arithmetical means between 11 and 32.
6. Insert 9 arithmetical means between and 30.
7. Find x, if are in A. P.
8. The 7th term of an arithmetical progression is 17, and the 13th term is 59. Find the 4th term.
9. How can you turn an A. P. into an equation?
10. Given find d and l.
11. Find the sum of the first n odd numbers.
12. An arithmetical progression consists of 21 terms. The sum of the three terms in the middle is 129; the sum of the last three terms is 237. Find the series. (Look up the short method for such problems.) (Mass. Inst. of Technology.)
13. B travels 3 miles the first day, 7 miles the second day, 11 miles the third day, etc. In how many days will B overtake A who started from the same point 8 days in advance and who travels uniformly 15 miles a day?
Reference: The chapter on Arithmetical Progression in any algebra.
GEOMETRICAL PROGRESSION
1. Define a geometrical progression.
Learn to derive the four formulas in geometrical progression:
2. How many terms must be taken from the series 9, 18, 36, ··· to make a total of 567?
3. In the G. P. 2, 6, 18, ···, which term is 486?
4. Find x, if are in geometrical progression.
5. How can you turn a G. P. into an equation?
6. Insert 4 geometrical means between 4 and 972.
7. Insert 6 geometrical means between and 5120.
8. Given find r and S.
9. If the first term of a geometrical progression is 12 and the sum to infinity is 36, find the 4th term.
10. If the series ··· be an A. P., find the 97th term. If a G. P., find the sum to infinity.
11. The third term of a geometrical progression is 36; the 6th term is 972. Find the first and second terms.
12. Insert between 6 and 16 two numbers, such that the first three of the four shall be in arithmetical progression, and the last three in geometrical progression.
13. A rubber ball falls from a height of 40 inches and on each rebound rises 40% of the previous height. Find by formula how far it falls on its eighth descent. (Yale.)
Reference: The chapter on Geometrical Progression in any algebra.
THE BINOMIAL THEOREM
1. Review the Binomial Theorem laws. (See Involution.)
Expand:
2.
3.
4.
5.
6.
7.
8.
Show by observation that the formula for the
9. Indicate what the 97th term of would be.
10. Using the expansion of in (8), derive a formula for the rth term by observing how each term is made up, then generalizing.
Using either the formula in (8) or (10), whichever you are familiar with, find:
11. The 4th term of
12. The 8th term of
13. The middle term of
14. The term not containing x in
15. The term containing in
Reference: The chapter on The Binomial Theorem in any algebra.
MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND
1. Solve the equation obtaining the values of the roots correct to three significant figures. (Harvard.)
2. Write the roots of (Sheffield Scientific School.)
3. Solve (Yale.)
4. Solve the equation for x, taking and and verify your result. (Harvard.)
5. Solve
6. Solve (Coll. Ent. Board.)
7. Find all values of x and y which satisfy the equations:
(Mass. Inst. of Technology.)
8. If and represent the roots of find and in terms of p, q, and r. (Princeton.)
9. Form the equation whose roots are and
10. Determine, without solving, the character of the roots of (College Entrance Board.)
11. If prove that (College Entrance Board.)
12. Given Prove that (Sheffield.)
13. The 9th term of an arithmetical progression is the 16th term is Find the first term. (Regents.)