Determine the prime-power representation of each of the following integers. (a) \( 59,930,332 \) (b) 48,400 (c) \( 1,882,384 \) (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. (f) Guess how you might know that from a glance at its prime-power representation that \( 27,543,608 \) is the cube of a natural number. Explain. (a) The prime-power representation of \( 59,930,332 \) is \( 2^{2} \cdot 7^{3} \cdot 11^{2} \cdot 19^{2} \). (Type your answer using exponential notation.) (b) The prime-power representation of 48,400 is \( \square \) (Type your answer using exponential notation.)

The prime-power representation of 48,400 is \( 2^{4} \cdot 5^{2} \cdot 11^{1} \). To check which numbers are squares, we can analyze their prime-power representations. A number is a perfect square if all the exponents in its prime-power representation are even. Therefore, \( 59,930,332 \) has exponents of \( 2, 3, 2, 2 \) (not all even, so not a square). For \( 48,400 \), \( 4, 2, 1 \) (again, not all even, so not a square). The number \( 1,882,384 \) has a prime-power representation of \( 2^{6} \cdot 3^{1} \cdot 7^{2} \) (not a square either since 1 is odd). For part (e), a perfect square in prime-power representation simply means that each prime factor's exponent is even. For part (f), \( 27,543,608 \)’s representation allows for quick identification as a cube if all the exponents are multiples of 3, making it easy to spot a potential cube by just looking at the numbers.