ALGEBRA - solved problems
  Radicals (roots) and/or exponentiation with fractional (rational) exponent
Rationalizing a denominator
75.    Rationalize given fractions.
Solutions:
 
        since   a3 - b3 = (a - b) (a2 + ab + b2)
Rationalizing a denominator is a method for changing an irrational denominator into a rational one.
Radicals and/or fractional (rational) exponents
76.    Solve given expressions.
Solutions:
 
77.    Simplify given expressions.
Solutions:
 
  Algebraic expressions
Simplifying algebraic expressions
78.    Simplify algebraic expressions.
Solutions:   a)   - 4a3 + 3a2 + 5a3 - 7a2 = (- 4 + 5) a3 + (3 - 7) a = a3 - 4a2,
b)   (x- x + 1) (x + 1) = x3 - x2 + x + x2  - x + 1 = x3 + 1.
Evaluating algebraic expressions
79.    Evaluate the expression  x- 6xy + 9y2  for x = 2  and  y -1.
Solution: x- 6xy + 9y2 = 2- 6 2 (- 1)  + 9 (-1)2 = 4 + 12 + 9 = 25.
Expanding algebraic expression by removing parentheses ( brackets)
80.    Expand given expressions.
Solutions:   a)   (a - b)2 = (a - b) (a - b) = a2 - ab - ab + b2 = a2 - 2ab + b2,
b)   (a - b) (a + b) = a2 - ab + ab - b2 = a2 - b2,
c)   (x + y) (x2 - xy + y2) = x3 - x2y + xy2 + x2y  - xy2 + y3 = x3 + y3.
The square of a binomial (or binomial square)
81.    Square given binomials.
Solutions:   a)   (a + b)2 = (a + b) (a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2,
b)   (2x + 3)2 = (2x)2 + 2 (2x) 3 + 32 = 4x2 + 12x + 9,
c)   (x - 2y)2 = x2  + 2 x (-2y) + (-2y)2 = x2 - 4xy + 4y2.
Squaring trinomial (or trinomial square)
82.    Square given trinomials.
Solutions:   a)  (x2 - 2x + 5)2 = (x2)2 + (2x)2 + 52 + 2 x2 (-2x) + 2 x2 5 + 2 (-2x) 5 =
                           = x4 + 4x2 + 25 - 4x3 + 10x2 - 20x = x - 4x3 + 14x2 - 20x + 25,
b)  (a3 - a2b - 3ab2)2 = (a3)2 + (a2b)2 + (3ab2)2 + 2a3 (-a2b) + 2a3 (-3ab2) + 2(-a2b) (-3ab2) =
  = a6 + a4b2 + 9a2b4 - 2a5b - 6a4b2 + 6a3b= a6 - 5a4b2 + 9a2b4 - 2a5b + 6a3b3.
Cube of a binomial
83.    Cube (rise to third power) given binomials.
Solutions:   a)  (a - b)3 = (a - b)2 (a - b) = (a2 - 2ab + b2) (a - b)
                  = a3 - 2a2b + ab2 - a2b + 2ab2 - b3 = a3 - 3a2b + 3ab2 - b3,
b)  (x - 2)3 = x3 + 3 x2 (-2) + 3 x (-2)+ (-2)3 = x- 6x2 + 12x - 8,
c)  (2x + y)3 = (2x)3 + 3 (2x)2 y + 3 (2x) y+ y3 = 8x3 + 12x2y + 6xy+ y3.
Solved problems contents
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.