# 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
2x2 - 32 + 32 = 0 + 32
Simplify
2x2 = 32
Divide both sides by 2
x2 = 16
Solve by extracting square root
x = ± √(16) = ± 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

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