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 - 12÷3×2 =
   Solution
   According to order of operations, 12÷3×2 (division and multiplication) is done first from left to right
   12÷3×2 = 4 × 2 = 8
   Hence
   30 - 12÷3×2 = 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 + 6×4÷3)
   Solution
   According to order of operations, inner brackets first where 6×4÷3 is first calculated since it has a multiplication and a division.
   6×4÷3 = 24÷3 = 8
   Hence
   11 + sqrt(- 4 + 6×4÷3) = 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?

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