Download Benchmark 1 Review with answers

Benchmark Review
Unit 1
1. What are the first 15 perfect squares?
1,4,9,16,25,36,49,64,81,100,121,144,169,196,225
2. List the first 6 perfect cubes.
1,8,27,64,125,216
3. Are there any numbers that are both perfect squares and perfect cubes?
1 and 64
4. Find the sum of all the integers between √3 and √50.
The integers that are between these 2 square roots are
√4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7 so we sum up 2+3+4+5+6+7 = 27
5. If you have a square picture that is 49 square inches large. How many 1 inch by 1 inch square
tiles would you need to make a square frame that would sit around the outside of the picture?
By taking the square root of 49 we find that each side of the picture is 7 inches long so a frame would
cover all 4 sides but then to connect the frame you must also use tiles in the corners so 7 times 4 is 28
plus the 4 corner tiles is 32 total tiles needed
6. How would you estimate √52? What do you estimate √52 to be?
The √52 lies between the perfect squares 49 and 64 so it must be between 7 and 8 since 52 is closer to
49 then we can estimate √52 to be about 7.2
7. Place the following numbers on a number line.
a. √12 should be around 3.4 on number line
b. √9 should be on 3 on number line
c. −√6 should be between -2 and -3 around -2.3 on number line
3
27
8. Simplify 5√125
3
3
9. Simplify √350 − 125 + √64
19
10. Find all possible solutions
a. √16
4, -4
3
b. √27 3
11. −√150 is between which 2 integers? Between -12 and -13
Unit 2 Review
12. What type of number are the following:
a. -2
integer, rational
b. ½
rational
c. √3
irrational
13. Give 3 examples of irrational numbers. Pi, and decimal that goes on forever without repeating,
any square root of a non perfect square
14. What is the definition of an irrational number? Any number that can not be written as a
fraction a/b where a and b are integers.
15. What is the definition of a rational number? Any number that can be written as a fraction a/b
where a and b are integers.
16. Convert the following to a fraction in simplest form
̅̅̅̅
a. .36
4/11
b. .8̅
8/9
̅
c. .45
41/90
17. How do you determine if a square root will be rational or irrational?
If the number under
the root is a perfect square it will be rational if not it will be irrational.
18. What type of number is represented by the following image?
the fraction 4/3.
Rational, this can be written as
19. Give 2 values x when used in the expression results in an irrational number. Give 2 values of x
that result in rational numbers. √−𝑥 + 15
If x is any value to where when you simplify under the root you get a perfect square then that
will be rational ex. -1,6,11,15…any number that does not simplify to a perfect square will be
irrational.
20. Convert 4/7 to a decimal.
.571428…repeating
Unit 3 Review
𝑥5
21. Identify the students mistake on this problem 𝑥 7 = 𝑥 12
The student added the exponent but should have subtracted to get x-2 then because the exponent was
negative should have move x-2 to the denominator and turned the exponent positive to give an answer
of 1/x2
22. How do you simplify the following expression? 𝑛−3 ∙ 𝑛4 ∙ 𝑛8
Add the exponents together and keep the base.
23. Simplify 5𝑥 2 (𝑥𝑦)6
5x8y6
24. Give 4 different representations of x72 using different rules of exponents.
Examples where multiplying with the same base should have exponents that add to equal 72.
Examples with an exponent in parenthesis being raised to an exponent outside the parenthesis
should multiply to equal 72. Exponents with division should subtract to equal 72.
25. What is the value of 2−5 ∙ 22
1/8
26. Simplify (𝑥 2 𝑦)−5 ∙ 𝑥 4 𝑦 6
𝑦
𝑥6
27. What is anything to the zero power? What type of exponents would give a value that is
smaller than something to the zero power?
Anything to the zero power equals 1. The only types of exponents that would give values smaller than 1
are negative exponents because they move the base to the denominator and become fractions for
1
instance 2-2 becomes 22 or ¼ which is smaller than 1.
−64𝑥 9
28. Simplify 8𝑥 3 𝑦−2
-8x6y2
29. What is the rule for raising an exponent to another exponent?
When raising an exponent to an exponent you multiply the exponents and keep the base.
30. Explain how to do this problem (4𝑥 4 𝑦)−2
Everything in the parenthesis gets taken to the -2 so you have 4-2x-8y-2, next you can’t have
negative exponents in your answer so all of the bases with the negative exponents need to be
moved. In this case they are all starting in the numerator so they move to the denominator as
1
42 𝑥 8 𝑦2
=
1
16𝑥 8 𝑦 2
31. What do you do when you get a negative exponent?
Negative exponents mean that your base is in the wrong place, so if the base is in the numerator
then move the base with the exponent to the denominator and then the exponent becomes
positive, if it is in the denominator then move the base with the exponent to the numerator and
then it becomes positive.
32. What is the difference between -22 and (-2)2 and why?
-22 means to square to first and then multiply by the negative therefore giving a final answer as
negative, however, -2 in parenthesis then squared means that everything in the parenthesis is squared
so negative times negative is positive giving a final answer of positive 4.
Unit 4 Review
33. The number 387,000,000,000 can be written as n x 1011 what is the value of n?
3.87
34. How do you order numbers from least to greatest in scientific notation?
First order the exponents from least to greatest. Second if one more than one of the exponents are the
same then order using the front numbers.
35. The mass of yacht is approximately 5.6 x 106 and the mass of a small fishing boat is
approximately 2.8 x 102 how many times larger is the yacht?
Divide 5.6 x 106/2.8 x 102 = 2 x 104
36. When determining how many times larger one object is over another what operation should
you use?
You would divide the numbers.
37. How do you multiply numbers in scientific notation?
Multiply the front numbers together, add the exponents, then convert to proper scientific notation if
necessary.
38. How do you divide numbers in scientific notation?
Divide the front numbers, subtract the exponents, then convert to proper scientific notation if
necessary.
39. A rocket travels at 105 meters per second. How many meters will it travel in 1020 seconds?
Multiply these two together so add the exponents since they are the same base 1025
40. Multiply (3.1 x 106)(5.9 x 1017)
(3.1 x 5.9)(106 x 1017) = 18.29 x 1023 = 1.829 x 1024
41. If I currently have 3.0 x 106 coins and want to increase that number by a third, how would I do
that and how much would I have now?
First find 1/3 of 3.0 x 106 which is 1.0 x 106 next we need to add 3.0 x 106 + 1.0 x 106 = 4.0 x 106
42.
298
re-write
100,000
in scientific notation
2.98 x 10-3