Solutions

Algebra Set #1

Solutions and comments in blue font

Problems to Do:

1. Simplify each to a single fraction or number:

 

a)                      70/431

 

b)      (2/3)3 ÷( (1 – 7/8)2 + ½ )    4096/6939

c)      (2*108)3        8x1024 

d)   (2*104)4 ÷ (4*102)2   1012

 

2.  Simplify each expression below as much as possible; your answer should have positive exponents only.

 

a.  (a3b-2)-3(a2b-2)(a-2b-1)-1   b5/a5           b)  (x-1 – y-1)-1(x-1y-1)-1  xy/(1/x – 1/y)    or also x2y2/(y-x)

 

c)   (x +y)-2(x +y)3(x2y3)-2  (x+y)/(x4y6)

 

d)     1/ (a2bc14)

 

3        Please simplify each of the following expressions as much as possible

 

a) 1252/3  25     b)   3-2   1/9        c)  45/2   32   d)  27-4/3  1/81         e)   (-1)5331  -1  f) 31252/5  25    (3125 = 55)

 

 

4.Given that 1 meter = 39 inches, what is the distance to the Sun in kilometers if the distance in miles is assumed to be 93 million?

 

Please show all steps leading to your answer and pay close attention to units.

 

1 mile = 5280  feet  =5280 x12 inches

                            =5280 x12/39 meters

so 93 million miles =  93 x106 x 5280 x12/39 meter

                           = (93 x 528 x 12 /39) x 107  meters

                           =  (15108.9231) x 107  m

                           =  1.511 x 1011  meters  (rounded)

          but 1000 meters = 1 kilometer = 1 km   so this is equal to  1.511 x 108  km 

 

5.  Is it true in general that

 

a)    (x y)n  = xnyn   ?     (n a positive integer)   YES

b)    (x + y )n  = xn + yn  ?     NO unless n =1         counterexample: (x +y)2 = (x+y)(x+y) = x2 + 2xy + y2   not x2 + y2

c)     1/x  +  1/y  =  1/(x+ y)   NO   counterexample:  1/2 + 1/5  =  7/10  not  1/7 

if your answer to any of them is no, provide a counterexample.