Exponents and Radicals
Exponents and Radicals
Objectives
Students will explore how powers are used to represent numbers, the properties of exponents, and the relationship between powers and roots. Students will:
- multiply and divide powers with the same base, and write the product or quotient as a power with the same base.
- represent large and small quantities using scientific notation.
- solve simple quadratic and cubic equations equivalent to x2 = n and x3 = n.
Essential Questions
- How is mathematics used to quantify, compare, represent, and model numbers?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- Simplifying Expressions with Exponents
https://www.purplemath.com/modules/simpexpo.htm - It’s the Law—The Laws of Exponents
http://oakroadsystems.com/math/expolaws.htm - Algebra – Using the rules of exponents to simplify (YouTube)
https://www.youtube.com/watch?v=asRk7gDWqAo - Scientific Notation. Practice converting between scientific notation and normal numbers
http://janus.astro.umd.edu/astro/scinote/ - Exponents: Scientific Notation
https://www.purplemath.com/modules/exponent3.htm - Scientific Notation
http://www.nyu.edu/pages/mathmol/textbook/scinot.html - Decimals to Scientific Notation
http://www.aaastudy.com/dec71ix2.htm - Solving Quadratic Equations: Solving by Taking Square Roots
https://www.purplemath.com/modules/solvquad2.htm - Cube and Cube Root Calculator
http://www.csgnetwork.com/cuberootcubecalc.html
Formative Assessment
-
View
Multiple-Choice Items:
- What is the value of 49 × 45?
A
44
B
49
C
414
D
445
- What is the value of 812 ÷ 8−3?
A
8−4
B
8−9
C
84
D
815
- What is the value of 512 ÷ 53 × 52?
A
56
B
57
C
58
D
511
- In scientific notation, expressed as a × 10n, what must be true about the factor a?
A
0 < a < 1
B
1 ≤ a < 10
C
1 ≤ a < 100
D
0 < a < 10
- What power of 10 represents the ten-thousandths place?
A
B
C
D
- What number does 3.642E19 represent on a calculator display?
Note that calculator displays vary.
A
.0000000000000000003642
B
69.198
C
36,420,000,000,000,000,000
D
22.642
- What are the solutions to the equation x2 = 61?
A
±√61
B
±30.5
C
±61
D
±122
- The solution to x3 = k is
. What is the value of k?
A
–125
B
–5
C
5
D
125
- If p is a positive integer, how many solutions does the equation x2 – p = 0 have?
A
0
B
1
C
2
D
p
Multiple-Choice Answer Key:
1. C
2. D
3. D
4. B
5. D
6. C
7. A
8. A
9. C
Short-Answer Items:
- Create an expression involving two powers that, when multiplied, has a value of 1280. Also, create an expression involving two powers that, when one is divided by the other, has a value of 640.
- The population of the United States is approximately 308 million people, and the population of the world is approximately 6.8 billion people. Write these numbers in scientific notation, and name the factor and exponent of each.
- Explain why an equation of the form x3 = n where n is a negative number has one solution and why an equation of the form x2 = n has no solution.
Short-Answer Key and Scoring Rubrics:
- Create an expression involving two powers that, when multiplied, has a value of 1280. Also, create an expression involving two powers that, when one is divided by the other, has a value of 640.
Sample Answers: 1240 × 1240 and 6100 × 660
Points
Description
2
The student creates two valid expressions that simplify to 1280 and 640, respectively.
1
The student creates one valid expression (either involving multiplication or division) that simplifies to 1280 or 640, but not both.
0
The student does not create any valid expression.
- The population of the United States is approximately 308 million, and the population of the world is approximately 6.8 billion. Write these numbers in scientific notation and name the factor and exponent of each.
The United States population is 3.08 × 108 , and the population of the world is 6.8 ×109 . The factors are 3.08 and 6.8, and the exponents are 8 and 9.
Points
Description
2
- The student’s work has correct numbers with no mathematical or computational errors.
- Factors and exponents are named correctly.
- Student demonstrates thorough understanding of scientific notation.
1
- The student’s work has some minor mathematical or computational errors.
- Scientific notation equations are correct, but the factors and exponents are incorrect.
- Student demonstrates partial understanding of scientific notation.
0
- Scientific notation equations are incorrect (the factor is not between 1 and 10).
- The factors and exponents are incorrectly identified.
- The student demonstrates no understanding of scientific notation.
12. Explain why an equation of the form x3 = n where n is a negative number has one solution and why an equation of the form x2 = n has no solution.
The first equation has a solution because when you multiply a negative number by itself three times, like –4 × –4 × –4, the answer is a negative number. The second equation has no solution because no matter how many times you multiply the equation by itself, you can never get a negative number.
Points
Description
2
The student provides valid, mathematical explanations regarding both the cubic and the quadratic equation.
1
The student provides valid, mathematical explanations regarding either the cubic or the quadratic equation, but not both.
0
The student does not provide any valid, mathematical explanations regarding either equation.
Performance Assessment:
Using the Internet or other sources, students should compile a report containing the lengths of/distances to at least 6 different objects, all expressed in scientific notation using the same unit. The list should contain at least 2 very small objects (cells, molecules, etc.) and 2 very large objects (planetary radii, distance between planets, etc.) and 2 others of their choice.
Students should make a poster or other presentation displaying the 6 items in increasing order of size. They should also determine how many times larger each item is than the smallest item in the list.
Finally, they should cut out a small circle with a radius of 1 inch and use that to represent the smallest length. Then, using ratios, they should determine what the radius of a
proportionally-sized circle would be, converting inches to feet, yards, miles, or any other appropriate unit.Performance Assessment Scoring Rubric:
Points
Description
4
- The student finds 6 items, at least 2 of which are small and 2 of which are large, and correctly lists their lengths / distances in scientific notation.
- The 6 items are correctly listed in increasing order.
- The poster correctly displays how many times larger each item is than the smallest item on the list.
- The student correctly determines the radius of a proportionally-sized circle, in the appropriate units, for each of the 5 larger objects.
- The student demonstrates an advanced understanding of scientific notation.
3
- The student completes 3 of the 4 tasks listed above.
- The student demonstrates some understanding of scientific notation.
2
- The student completes 2 of the 4 tasks listed above.
- The student demonstrates limited understanding of scientific notation.
1
- The student completes 1 of the 4 tasks listed above.
- The student demonstrates very limited understanding of scientific notation.
0
- The student does not complete any of the tasks listed above.
- The student demonstrates no understanding of scientific notation.
Final 06/28/13
