8th Grade expressions and equations
Exponential Expressions
1. Simplify the following expression.
/ A. // B. /
/ C. /
/ D. /
Scientific Notation
2. Which of the following would be a reasonable estimate for the weight of a pencil?
/ A. / 1 × 10-20lb/ B. / 1 × 10-2lb
/ C. / 1 × 1020lb
/ D. / 1 × 102lb
Scientific Notation
3. The Earth is approximately 2 × 107 km from Venus at its closest approach. The Earth is also about 4 × 1013 km from Proxima Centauri, a star. Which of the following is true?
/ A. / The distance from Earth to Proxima Centauri is approximately two trillion times the distance from Earth to Venus./ B. / The distance from Earth to Proxima Centauri is approximately two billion times the distance from Earth to Venus.
/ C. / The distance from Earth to Proxima Centauri is approximately two thousand times the distance from Earth to Venus.
/ D. / The distance from Earth to Proxima Centauri is approximately two million times the distance from Earth to Venus.
Exponential Expressions
4. Simplify the following expression.
(65)2 × 66
/ A. / 613/ B. / 616
/ C. / 64
/ D. / 642
Systems of Equations
5. The following system of equations is graphed below.
Find the solution to the system.
/ A. / x = 3, y = -3/ B. / x = 0, y = -1
/ C. / x = 3, y = 4
/ D. / x = -3, y = 1
Systems of Equations
6.
-2x + 4y / = / -4x - 3y / = / 6
The system of equations above is graphed below. Find the solution to the system.
/ A. / x = 0, y = -6/ B. / x = -6, y = -4
/ C. / x = -6, y = -3
/ D. / x = -6, y = 4
Proportional Relationships
7. Joanna is filling her swimming pool with water. The graph below shows the depth of the water as time passed.
At what rate is the depth of the pool water increasing?
/ A. // B. /
/ C. /
/ D. /
Solving Linear Equations
8. Which value of x makes the following equation true?
3(x - 3) + 65 = -4x
/ A. // B. /
/ C. /
/ D. /
Square and Cube Roots
9. Simplify:
/ A. / 11/ B. / 12
/ C. / 10
/ D. / 9
Square and Cube Roots
10. Which of the following best describes the solution to the equation below?
Note: Only consider the positive solution.
x2 = 2
/ A. / It is greater than zero but less than one./ B. / It is an irrational number.
/ C. / It is a fraction.
/ D. / It is a repeating decimal.
Exponential Expressions
11. Simplify the following expression.
/ A. // B. /
/ C. /
/ D. /
Square and Cube Roots
12. Determine which of the following are the solutions to the equation below.
/ A. // B. /
/ C. /
/ D. /
Proportional Relationships
13. Katherine is having cloth napkins embroidered with her initials. The cost for the embroidery is $4.50 per napkin.
Which graph represents the total cost for embroidering xnapkins?
W. /
X.
Y. /
Z.
/ A. / Z
/ B. / Y
/ C. / W
/ D. / X
Solving Linear Equations
14. Which of the following best describes the solution to the equation below?
28x + 24 = -42x - 8
/ A. / infinite real solutions/ B. / exactly one real solution, /
/ C. / no real solutions
/ D. / exactly one real solution, /
Proportional Relationships
15. Philip wants to replace his existing fence. He received a quote from a general contractor based on the graph below. The graph shows the cost of a new cedar fence based on the number of feet.
What is the unit rate of the graph?
/ A. / $20 per foot/ B. / $10 per foot
/ C. / $5 per foot
/ D. / $1 per foot
Answers
1. C
2. B
3.D
4.B
5.D
6.B
7.D
8.D
9.C
10.B
11.D
12.D
13.A
14.B
15. A
Explanations
1. When dividing two terms with the same base, subtract the exponents.
Simplify the expression.
When the exponent is negative, the expression is equivalent to its reciprocal with a positive exponent.
Simplify the expression.
2. Consider each option.
1 × 10-20lb / = / 0.00000000000000000001 lb1 × 10-2lb / = / 0.01 lb
1 × 102lb / = / 100 lb
1 × 1020lb / = / 100,000,000,000,000,000,000 lb
The option 1 × 10-20lb would be too small for the weight of a pencil.
The options 1 × 102lb and 1 × 1020lb are too large for the weight of a pencil.
Therefore, 1 × 10-2lb is the best estimate for the weight of a pencil.
3. Compare the distance from Earth to Proxima Centauri to the distance from Earth to Venus by dividing the distance to Proxima Centauri by the distance to Venus.
Since 2 × 106 is equal to 2,000,000, the distance from Earth to Proxima Centauri is approximately two million times the distance from Earth to Venus.
4. Whenever an exponent is raised to a power, multiply the exponent and the power together.
(am)n / = / a(m × n)(65)2 / = / 6(5 × 2)
= / 610
When multiplying two or more exponential expressions with the same base, add the exponents.
am × an / = / a(m + n)610 × 66 / = / 6(10 + 6)
= / 616
5. The point of intersection of the two graphs gives the solution to the system of equations.
The two lines intersect at (-3, 1).
So, x = -3 and y = 1 is the solution.
6. The solution to the system of equations is the point where the two lines intersect.
The two lines intersect at the point (-6, -4). Therefore, x = -6 and y = -4 is the solution to the system.
7. To determine the rate of feet per hour, use two sets of coordinates from the line. Use the points (0, 0) and (2, 3).
Divide the difference in the y-coordinates by the difference in the x-coordinates to determine the rate.
Therefore, the rate is shown below.
8. Isolate x on one side of the equation to solve.
3(x - 3) + 65 / = / -4x3x - 9 + 65 / = / -4x
3x + 56 / = / -4x
3x + 56 - 3x / = / -4x - 3x
56 / = / -7x
56 ÷ -7 / = / -7x ÷ -7
-8 / = / x
9. When a number is written inside a radical sign (), it is read as "the square root of" that number.
Since 102 = 100, the square root of 100 is 10.
10. The solution to the equation x2 = p is a rational number when p is a perfect square.
In this case, p = 2. Since 2 is not a perfect square, the solution to the equation is not rational.
Therefore, it is an irrational number.
11. When multiplying two terms with the same base, add the exponents.
Simplify the expression.
When the exponent is negative, the expression is equivalent to its reciprocal with a positive exponent.
Simplify the expression.
12. Find the numbers that equal 5 when squared.
Therefore, the solutions to the equation are .
13. The total cost for embroidering x napkins can be represented by the following equation.
y = $4.50x
The slope of the line, $4.50, represents the cost per napkin.
The y-intercept is 0.
The graph that represents this situation is graph Z.
14. Solve the given equation.
Therefore, the solution to the equation 28x + 24 = -42x - 8 is shown below.
exactly one real solution, /15. The unit rate, or slope, can be determined from two points on the graph.
For example, the points (20, 400) and (60, 1,200) can be used to find the unit rate.
Therefore, the unit rate of the graph is $20 per foot.