ALGEBRA - solved problems
Adding and subtracting same powers
60.    Add or subtract powers.
Solutions: a)  2a2 + 5a - a2 - a = a2 + 4a,       b)  3a4 - 7a4 + a4 = (3 - 7 + 1) a4  = -3a4
Only the same powers (with same base and the exponent) can be added or subtracted.
We also say that only like or similar terms can be added. The number that multiply the power is called coefficient. We add or subtract the same powers by adding or subtracting their coefficients.
The rules for powers (or exponents)
Rules Examples
 am a = am + n  a)  a3 a4 = a3 + 4,                 b)  24 25 = 24 + 5 = 29 = 512, 
 am a = am - n  a)  x5 x3  = x5 - 3  = x2,       b)  0.14 0.13 = 0.14 - 3 = 0.1, 
 an b = (a b)n  a)  24 34 = (2 3)4 = 64,        b)  45 0.85 = (4 0.8)5 = 3.25,
 an b = (a b)n  a)  x6 y6  = (x y)6,             b)  125 35 = (12 3)5 = 45,
   
 (am) = am n,  a)  (a3)4 = a3 4 = a12,             b)  (45)3 = 45 3 = 415, 
   
   
   
Simplifying an exponential expression
61.    Simplify given expressions.
Solutions:  
 
Scientific notation
62.    Write in scientific notation and vice versa.
Solutions: a)  0.0000007054 = 7.054 10-7,         b)  5.2 10-4 = 0.00052
  c)  4507000000 = 4.507 109,             d)  1.04 105 = 104000
A number in scientific notation is written as the product of a number called coefficient and a power of 10.
While converting to scientific notation, the decimal point of the coefficient is placed behind the first non-zero digit. The sign of the exponent of the power indicates, how many places the decimal point was moved to the right or to the left.
  Quadratic equation  x2 = aa > 0
63.    Solve quadratic equations.
Solutions: a)   x2 = 16   b)   (x - 5)2 = 49
        x2 - 42 = 0         (x - 5)2 - 72 = 0
       (x - 4) (x + 4) = 0         [(x - 5) - 7] [(x - 5) + 7] = 0
        x - 4 = 0    =>   x1 = 4         (x - 5) - 7 = 0    =>   x1 = 12
        x + 4 = 0    =>   x2 = - 4         (x - 5) + 7 = 0    =>   x2 = -2
64.    Solve quadratic equations.
Solutions:
 
  Square root
Properties of square roots
Properties Examples
       
       
       
         
         
         
         
Adding, subtracting, multiplying and dividing square roots
65.    Solve given square roots operations.
Solutions:  
   
   
   
   
   
Rationalizing a denominator
66.    Eliminate radicals from denominators of fractions .
Solutions:  
   
   
Solved problems contents
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