[ ] D) \(3^2 \times 5\)
7. Which expression shows the prime factorization of 180?
[ ] A) \(4 \times 9 \times 5\)
[ ] B) \(2 \times 3^2 \times 10\)
[ ] C) \(2^2 \times 3^2 \times 5\)
[ ] D) \(2^3 \times 3 \times 5\)
8. Which number has the prime factorization \(2^3 \times 3 \times 7\)?
[ ] A) 56
[ ] B) 126
[ ] C) 168
[ ] D) 336
9. Which of the following expressions is a valid prime factorization?
[ ] A) \(2^3 \times 5^2 \times 11\)
[ ] B) \(2 \times 3 \times 15\)
[ ] C) \(2^4 \times 3^2 \times 1\)
[ ] D) \(2^2 \times 4 \times 7\)
10. Elena factored 96 and wrote: "The prime factorization of 96 is \(2^3 \times 12\)." What error did Elena make?
[ ] A) \(2^3 \times 12\) does not equal 96.
[ ] B) 12 is composite; it must factor to \(2^2 \times 3\).
[ ] C) Elena cannot use exponents on prime bases.
[ ] D) 96 is prime and has no factors.
25 Points Total
11. Evaluate each square root. (2 pts each = 12 pts)
\(\sqrt{49}\) Ans:
\(\sqrt{100}\) Ans:
\(\sqrt{121}\) Ans:
\(\sqrt{169}\) Ans:
\(\sqrt{196}\) Ans:
\(\sqrt{225}\) Ans:
12. Solve for the positive unknown value. Show one step of reasoning. (3 pts each = 6 pts)
a) \(x^2 = 144\)
Take square root of both sides:
\(x =\)
b) \(y^2 = 256\)
Take square root of both sides:
\(y =\)
13. Real-World Application Problem: (7 pts total)
A landscape architect is designing a square courtyard with an area of \(225\text{ square meters}\).
a. Side length (\(s\)) of courtyard? (3 pts)
\(s =\) ______ meters
b. Perimeter border stone needed? (4 pts)
Perimeter = ______ meters
Prime Power Test • Page 2 of 2 Subtotal Points Page 2: _____ / 50
Exponential Form (2 pts): \(2^2 \times 3^3\)
Tree branching can vary (e.g., \(2 \times 54\)); award 9 pts for valid branch breakdown.
Prime Power Answer Key • Page 1 of 2 Subtotal Points Page 1: 50 / 50
Part II • Multiple Choice & Roots Key
Page 2: 50 Points Total
25 Points (5 pts each)
6. Prime Factorization of 75 CORRECT: B
• \(3 \times 5^2\) (\(3 \times 25 = 75\))
Distractor analysis: A and C contain composite numbers 25 and 15. D equals 45.
7. Prime Factorization of 180 CORRECT: C
• \(2^2 \times 3^2 \times 5\) (\(4 \times 9 \times 5 = 180\))
Distractor analysis: A and B contain composite factors 4, 9, 10. D equals 120.
8. Match \(2^3 \times 3 \times 7\) CORRECT: C
• \(168\) (\(8 \times 3 \times 7 = 24 \times 7 = 168\))
Distractor analysis: A is \(2^3 \times 7\). B is \(2 \times 3^2 \times 7\). D is \(2^4 \times 3 \times 7\).
9. Valid Prime Factorization CORRECT: A
• \(2^3 \times 5^2 \times 11\) (all bases 2, 5, 11 are prime)
Distractor analysis: B has 15 (composite). C has 1 (not prime). D has 4 (composite).
10. Elena's Factorization Error of 96 CORRECT: B
12 is composite; it must factor to \(2^2 \times 3\), giving the true prime factorization: \(2^5 \times 3\).
Note: \(2^3 \times 12\) does mathematically equal 96, but it is not a complete prime factorization.
25 Points Total
11. Square Root Evaluations (2 pts each = 12 pts)
\(\sqrt{49}\) = 7
\(\sqrt{100}\) = 10
\(\sqrt{121}\) = 11
\(\sqrt{169}\) = 13
\(\sqrt{196}\) = 14
\(\sqrt{225}\) = 15
12. Solving Quadratic Roots (3 pts each = 6 pts; 1 pt work + 2 pts answer)
a) \(x^2 = 144\)
Work: \(x = \sqrt{144}\)
Answer: \(x = 12\)
b) \(y^2 = 256\)
Work: \(y = \sqrt{256} = \sqrt{16 \times 16}\)
Answer: \(y = 16\)
13. Real-World Application Problem (7 pts total)
a. Side Length (\(s\)) (3 pts)
Formula: \(s = \sqrt{\text{Area}} = \sqrt{225}\)
\(s = 15\text{ meters}\)
(Award 1 pt for formula/work, 2 pts for correct value and units).
b. Perimeter Border Stone (4 pts)
Formula: \(\text{Perimeter} = 4s = 4(15\text{ m})\)
Perimeter = 60 meters
(Award 2 pts for multiplication by 4, 2 pts for correct total and units).
• 51 and 91: Most common missed primes. Remind students of the sum of digits test (\(5+1=6\)) and divisibility by 7 (\(7 \times 13 = 91\)).
• Tree Termination: Deduct 1 pt if a student stops branching before fully reaching prime numbers (e.g., leaving a 4, 6, 8, or 9 un-factored).
• Square Roots: Check for division by 2 confusion (e.g., writing \(\sqrt{100} = 50\) instead of \(10\)). Require justification if needed.
Prime Power Answer Key • Page 2 of 2 Total Test Score: 100 / 100
