Part 36

ALGEBRA A

TIME: TWO HOURS

1. Simplify [m + 1/m]^2 + [n + 1/n]^2 + [mn + 1/mn]^2 - [m + 1/m][n + 1/n][mn + 1/mn].

2. Find the prime factors of (a) (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3. (b) (2x + a - b)^4 - (x - a + b)^4.

3. (a) Simplify [(x^q)/(x^r)]^(q + r) [(x^r)/(x^p)]^(r + p)[{x^p/{x^q}]^(p + q).

(b) Show that ([^[1/(n+1)]]^(1/n))/([^[1/(n+2)]]^[1/(n+1)]) = {x^(1/n) . ^[1/(n+2)]}/{[x^2]^[1/(n+1)]}.

4. Define homogeneous terms.

For what value of n is x^n y^(5 - n/2) + x^(n + 1) y^(2n - 6) a homogeneous binomial?

5. Extract the square root of x(x - 2^(1/2))(x - 8^(1/2))(x - 18^(1/2)) + 4.

6. Two vessels contain each a mixture of wine and water. In the first vessel the quantity of wine is to the quantity of water as 1 : 3, and in the second as 3 : 5. What quantity must be taken from each, so as to form a third mixture which shall contain 5 gallons of wine and 9 gallons of water?

7. Find a quantity such that by adding it to each of the quantities a, b, c, d, we obtain four quantities in proportion.

8. What values must be given to a and b, so that (3a + 2b + 17)/2, (2a - 3b + 25)/3, 4 - 5a - 13b may be equal?

~MOUNT HOLYOKE COLLEGE~

ELEMENTARY ALGEBRA

TIME: TWO HOURS

1. Factor the following expressions:

(a) a^(3/4) - b^(3/4).

(b) x^2 y^2 z^2 - x^2 z - y^2 z + 1.

(c) 16(x + y)^4 - (2x - y)^4.

2. (a) Simplify

(a^2 + b^2){(b^4)/(b^2 - a^2) - a^2}/{a/(a + b) + b/(a - b)}}.

(b) Extract the square root of x^4 - 2x^3 + 5x^2 - 4x + 4.

3. Solve the following equations:

(a) 1/x + 1/y = 5, 1/(x^2) + 1/(y^2) = 13.

(b) x^2 - 5x + 2 = 0.

(c) [27x + 1]^(1/2) = 2 - 3[3x^(1/2)].

4. Simplify:

(a) 7^(1/3) + 256^(1/6) + [432/(-250)]^(1/3).