Determine the prime-power representation of each of the following integers. \( \begin{array}{lll}\text { (a) } 59,930,332 & \text { (b) } 48,400 & \text { (c) } 1,882,384 \\ \text { (d) Which numbers in parts } & \text { (a), (b), and (c) are squares? } \\ \text { (e) What can you say about the prime-power representation of a square? Explain briefly. } \\ \text { (f) Guess how you might know that from a glance at its prime-power representation that } \\ 27,543,608 \text { is the cube of a natural number. Explain. } \\ \text { (a) The prime-power representation of } 59,930,332 \text { is } 2^{2} \cdot 7^{3} \cdot 11^{2} \cdot 19^{2}\end{array} \) (Type your answer using exponential notation.) (b) The prime-power representation of 48,400 is \( 2^{4} \cdot 5^{2} \cdot 11^{2} \) (Type your answer using exponential notation.) (c) The prime-power representation of \( 1,882,384 \) is \( 2^{4} \cdot 7^{6} \). (Type your answer using exponential notation.) (d) Which numbers in parts (a), (b), and (c) are squares? Choose all that apply. A. \( 1,882,384 \) B. 48,400 \( \square \) C. \( 59,930,332 \) D. None of the numbers in parts (a), (b), or (c) are squares. D. (b) (b)

(a) We have \[ 59{,}930{,}332 = 2^{2} \cdot 7^{3} \cdot 11^{2} \cdot 19^{2}. \] (b) We have \[ 48{,}400 = 2^{4} \cdot 5^{2} \cdot 11^{2}. \] (c) We have \[ 1{,}882{,}384 = 2^{4} \cdot 7^{6}. \] (d) A number is a perfect square precisely when each exponent in its prime-power representation is even. Examining the factorizations: - For (a): the exponent of \(7\) is \(3\) (odd), so it is not a square. - For (b): all exponents (\(4\) for \(2\), \(2\) for \(5\), and \(2\) for \(11\)) are even, so it is a square. - For (c): both exponents (\(4\) for \(2\) and \(6\) for \(7\)) are even, so it is a square. Thus, the numbers in parts (b) and (c) are squares. (e) In the prime-power representation of a perfect square, every exponent is even. This is because if a number has the factorization \[ n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}, \] then \[ n^2 = p_1^{2a_1} p_2^{2a_2} \cdots p_k^{2a_k}. \] Thus, squaring any natural number results in all exponents being even. (f) A number is a perfect cube if and only if each exponent in its prime-power representation is a multiple of \(3\). When you see a prime factorization where every exponent is divisible by \(3\), you can immediately conclude that the number is the cube of another natural number. For example, if \[ 27{,}543{,}608 = 2^{a} \cdot 3^{b} \cdot 5^{c} \cdots \] with \(a\), \(b\), \(c\), … all divisible by \(3\), then \[ 27{,}543{,}608 = \left(2^{a/3} \cdot 3^{b/3} \cdot 5^{c/3} \cdots\right)^3. \] Thus, a glance at its prime-power representation showing exponents all multiples of \(3\) lets you know that \(27{,}543{,}608\) is a perfect cube.