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.
 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) · a2  = a3 - 4a2, b)   (x2  - x + 1) · (x + 1) = x3 - x2 + x + x2  - x + 1 = x3 + 1.
Evaluating algebraic expressions
 79 Evaluate the expression  x2  - 6xy + 9y2  for x = 2  and  y =  -1.
 Solution: x2  - 6xy + 9y2 = 22  - 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 = x4  - 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 + 6a3b3  = 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  + (-2)3 = x3  - 6x2 + 12x - 8, c)  (2x + y)3 = (2x)3 + 3 · (2x)2 · y + 3 · (2x) · y2  + y3 = 8x3 + 12x2y + 6xy2  + y3.
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