Solutions and Explanations to Intermediate Algebra Questions in Sample 1

Solutions and full explanations of intermediate algebra questions in sample 1 are presented.

  1. Write 230,000,000,000 in scientific notation.
    Solution
    Write the given number in the form
    a 10 n , where a is a real number such that 1 ≤ |a| < 10 and n is an integer.
    230,000,000,000 = 2.3 100,000,000,000 = 2.3 10 11

  2. Evaluate: 30 - 1232 =
    Solution
    According to order of operations, 1232 (division and multiplication) is done first from left to right
    1232 = 4 2 = 8
    Hence
    30 - 1232 = 30 - 8 = 22

  3. Evaluate: |4 - 8(3 - 12)| - |5 - 11| =
    Solution
    According to order of operations, inner brackets first. Hence
    |4 - 8(3 - 12)| - |5 - 11| = |4 - 8*(-9)| - |5 - 11|
    According to order of operations, multiplication within absolute value signs (which may be considered as brackets when it comes to order of operations) next. Hence
    = |4 + 72| - |5 - 11|
    = |76| - |-6|
    = 76 - 6 = 70

  4. Evaluate: -18 + 4(6 2)2
    Solution
    According to order of operations, inner brackets first. Hence
    -18 + 4(6 2)2 = -18 + 4(3)2
    According to order of operations, power next. Hence
    = -18 + 4*9
    According to order of operations, multiplication next. Hence
    = -18 + 36
    = 18

  5. Evaluate: 11 + sqrt(- 4 + 643)
    Solution
    According to order of operations, inner brackets first where 643 is first calculated since it has a multiplication and a division.
    643 = 243 = 8
    Hence
    11 + sqrt(- 4 + 643) = 11 + sqrt(- 4 + 8)
    = 11 + sqrt(4) = 11 + 2 = 13

  6. Simplify: 12x3 - 3(2x3 + 4x -1) - 5x + 7
    Solution
    First expand the term - 3(2x3 + 4x -1)
    12x3 - 3(2x3 + 4x -1) - 5x + 7 = 12x3 - 6 x3 - 12 x + 3 - 5x + 7
    Group like terms
    = 6 x3 - 17 x + 10

  7. Simplify:$(\dfrac{x^4}{x^3})^3$
    Solution
    Use quotient of powers formula $\dfrac{x^m}{x^n}=x^{m-n}$ to simplify $\dfrac{x^4}{x^3}$.
    $(\dfrac{x^4}{x^3})^3=(x^{4-3})^3=x^3$

  8. Simplify: $\dfrac{(3x^2y^{-2})^3}{(9xy^3)^3}$
    Solution
    Use power of quotient formula $\dfrac{a^m}{b^m}=(\dfrac{a}{b})^m$
    $\dfrac{(3x^2y^{-2})^3}{(9xy^3)^3}= (\dfrac{3 x^2y^{-2}}{9 x y^3})^3$
    $= (\dfrac{x^{2-1}}{3y^{3+2}})^3$
    $= (\dfrac{x}{3y^5})^3 $
    $= \dfrac{x^3}{27y^{15}}$

  9. Simplify: $\dfrac{(2x^{-3}y^4)^3(x^3 + y)^0}{(4xy^{-2})^3}$
    Solution
    Note that the above expression is defined when neither $x$ nor $y$ is equal to zero and therefore $(x^3 + y)^0 = 1$. Hence
    $\dfrac{(2x^{-3}y^4)^3(x^3 + y)^0}{(4xy^{-2})^3} = \dfrac{(2x^{-3}y^4)^3}{(4x y^{-2})^3} $
    $=(\dfrac{2x^{-3}y^4}{4x y^{-2}})^3$
    $= ((1/2) \dfrac{y^{4+2}}{x^{1+3}})^3 $
    $= (1/8) (\dfrac{y^6}{x^4})^3$
    $= (1/8) \dfrac{y^{18}}{x^{12}}$

  10. Write as a mathematical inequality:"9 is less than the product of M and N".
    Solution
    9 < M N

  11. Find the slope of the line perpendicular to the line y = (1/3)x - 7
    Solution
    Two lines are perpendicular if the product of their slopes is equal to -1. The slope of the given line is equal to 1 / 3. If m is the slope of the line perpendicular to the given line, then
    m (1/3) = -1
    Solve for m
    m = - 3

  12. Write an equation of the line with slope -3 and y-intercept (0 , -5).
    Solution
    y = m x + b is the general form of the equation of a line in slope intercept form. Hence for m = -3 and b = -5, we have the equation
    y = - 3 x - 5

  13. Solve the equation: -5 x + 20 = 25
    Solution
    Subtract 20 to both sides
    -5 x + 20 - 20 = 25 - 20
    Simplify
    - 5 x = 5
    Divide both side s b y -5
    x = - 1

  14. Solve the inequality: -3x + 4 < -8
    Solution
    Subtract 4 from both sides
    - 3 x + 4 - 4 < - 8 - 4
    Simplify
    - 3 x < - 12
    Divide both sides b y - 3 and change symbol of inequality
    x > 4

  15. Solve the equation: 2x2 - 32 = 0
    Solution
    Add 32 both sides
    2x2 - 32 + 32 = 0 + 32
    Simplify
    2x2 = 32
    Divide both sides by 2
    x2 = 16
    Solve by extracting square root
    x = ~+mn~ √(16) = ~+mn~ 4 , two solutions

  16. Solve the equation: -0.25x + 1.3 = -0.55x - 0.2

  17. Solve the equation: -0.25x2 + 1.5 = -10.75

  18. What is the slope of a line perpendicular to the line x = -3?

  19. What is the slope of a line parallel to the line x = 5?

  20. What is the slope of a line perpendicular to the line y = 6?

Answers to the Above Questions
  1. 2.3 1011
  2. 22
  3. 70
  4. 18
  5. 13
  6. 6x3 -17x + 10
  7. x3
  8. (1/27)(x3 / y15)
  9. (1/8)(y18 / x12)
  10. 9 < M N
  11. -3
  12. y = -3x - 5
  13. -1
  14. x > 4
  15. -4 , 4
  16. -5
  17. -7 , 7
  18. 0
  19. undefined
  20. undefined

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