A Review of Algebra

Name: A Review of Algebra
Author: Rivenburg, Romeyn Henry

Rivenburg, Romeyn Henry

Algebra -- Problems, exercises, etc.

Any two of the quantities x + y, x^2 + y^2, xy, x^3 + y^3,
        x^3 - y^3, x - y, x^2 +- xy + y^2, etc., given.
            x^2 + y^2 = 41,
            x + y = 9.

    CASE IV.
        Both equations symmetrical or symmetrical except for sign.
        Usually one equation of high degree, the other of the first
        degree.
            x^5 + y^5 = 242,
            x + y = 2.

    CASE V. Special Devices

        I. Solve for a compound unknown, like xy, x + y, (1)/(xy),
        etc., first.
            x^2y^2 + xy = 6,
            x + 2y = -5.

        II. Divide the equations, member by member.
            x^4 - y^4 = 20,
            x^2 - y^2 = 5.

        III. Eliminate the quadratic terms.
            4x + 3y = 2xy,
            7x - 5y = 5xy.

~Ratio and Proportion~

    Proportionals
        mean,
        third,
        fourth.

    Theorems
        1. Product of means equals product of extremes.
        2. If the product of two numbers equals the product of two
           other numbers, either pair, etc.
        3. Alternation.
        4. Inversion.
        5. Composition.
        6. Division.
        7. Composition and division.
        8. In a series of equal ratios, the sum of the antecedents
           is to the sum of the consequents as any antecedent, etc.

    Special method of proving four quantities in proportion. Let
    a/b = x, a = bx, etc.

~Progressions~

    Development of formulas.
                                       { l = ar^(n - 1).
        { l = a + (n - 1)d.            { S = (ar^n - a)/(r - 1).
        { S = (n/2)(a + l).            { S = (rl - a)/(r - 1).
        { S = (n/2)[2a + (n - 1)d].    { S[infinity] = (a)/(1 - r).

    Insertion of means
        Arithmetical.
        Geometrical.

~Binomial Theorem~

    Review of binomial theorem laws. See Involution.

    Expansion of (a + b)^n.

    Finding any term by
        key number method.
        r^(th) or (r + 1)^(th) term method.




A REVIEW OF ALGEBRA




ORDER OF OPERATIONS, EVALUATION, PARENTHESES


    Order of operations:
      First of all, raising to a power and extracting a root.
      Next, multiplication and division.
      Last of all, addition and subtraction.


Find the value of:

  1. 5 . 2^2 - 25^(1/2) / 5 + 2^2 . 8 / 4 - 2.

  2. (3 x 6 / 9)/2 - 2[100^(1/2)] / 5 + 4 . 2^3 - (14 . 2)/28.

  3. 9 . 2 / 6 + 3 - 2 . 4^2 / 8^(1/3) - 4 + (3 . 2^2)/6.

Evaluate:

  4. (a^4 - a^3 + b^3)/([a^2 b^2]^(1/2)) + (c[a^(1/2)] + a^3bc)/(abc),
     if a = 1, b = 2, c = 3.

  5. t^(1/3) + [tm]^(1/3)  +  m^(1/3), if t = 8, m = 27.

  6. (2[3 + 2d + a]^(1/2))/(3[a + b - cx - c]^(1/2))
     + ((3c - d)x)/(7ad - [abc]^(1/2)), if
     a = 5, b = 3, c = -1, d = -2, x = 0.

  7. a - {5b - [a - (3c - 3b) + 2c - 3(a - 2b - c)]},
     if a = -3,  b = 4, c = -5.                          (_Yale._)

Simplify:

  8. m - [2m - {3r - (4r - 2m)}].

  9. 2a - [5d + {3c - (a + [2d - 3a + 4c])}].

  10. 3c^2 + c(2a - [6c - {3a + c - 4a}]).

Public-domain text, read in full here on John Shaqi.

A Review of Algebra

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