This Question: 1 pt 18 of 24 (21 complete) Determine the prime-power representat

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This Question: 1 pt 18 of 24 (21 complete) Determine the prime-power representation of each of the following integers (a) 28,056,028 (b) 132,496 (c) 73,530,625 (d) Which numbers in parts (a), (b), and (c) are squares? (e) What can you say about the prime-power representation of a square? Explain briefly D Guess how you might know that from a glance at its prime-power representation that 45,882,712 is the cube of a natural number. Explain (a) The prime-power representation of 28,056,028 is Type your answer using exponential notation.) (b) The prime-power representation of 132,496 is Type your answer using exponential notation.) (e) The prime-power representation of 73,530,625 is Type your answer using exponential notation.) ad (d) Which numbers in parts (a), (b), and (c) are squares? Choose all that apply. A 28,056028 B. 132,496 . 73530,625 D. None of the numbers in parts (a). (b), or (c) are squares. (e) What can you say about the prime-power representation of a square? Explain briefy. O A. The exponents of each of the prime factors must be greater than two O B. The exponents of each of the prime factors must be less than or equal to two O C. The exponents of each of the prime factors must be two O D. The exponents of each of the prime factors in the prime-power representation are al evern 1) Guess how you might know from a glance at its prime-power representation that 45,882,712 is the cube of a natural number Explain. Click to select your answerls. MacBook Air 4

Explanation / Answer

a) The prime power factorization of number 28,056,028

2 x 2 x 7 x 7 x 7 x 11 x 11 x 13 x 13

Hence, the number can be written as 2^2 * 7^3 * 11^2 * 13^2

b)

The prime power factorization of number 132,496

2 x 2 x 2 x 2 x 7 x 7 x 13 x 13

Hence, the number can be written as 2^4 * 7^2 * 13^2

c)

The prime power factorization of number 132,496

5 x 5 x 5 x 5 x 7 x 7 x 7 x 7 x 7 x 7

Hence, the number can be written as 5^4 * 7^6

d) The correct answer is Option B and Option C

A is not a perfect square, since the power of 7 is an odd number

e) The correct answer is Option D

The exponents of each of the prime factors must be an even number, otherwise there will be a square root term coming when we will take the square root to check if the number is a perfect square

f) The Prime Factorization is:
2 x 2 x 2 x 179 x 179 x 179

Since the power of 2 and 179 are 3, hence the number is a perfect cube of 2 * 179 = 358

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