Determine the prime-power representation of each of the following integers. \( \begin{array}{lll}\text { (a) } 59,930,332 & \text { (b) } 48,400 & \text { (c) } 1,882,384 \\ \text { (d) Which numbers in parts } & \text { (a), (b), and (c) are squares? } \\ \text { (e) What can you say about the prime-power representation of a square? Explain briefly. } \\ \text { (f) Guess how you might know that from a glance at its prime-power representation that } \\ 27,543,608 \text { is the cube of a natural number. Explain. }\end{array} \) (a) The prime-power representation of \( 59,930,332 \) is (Type your answer using exponential notation.)

\[ 59,\!930,\!332 = 2^2 \cdot 7^3 \cdot 11^2 \cdot 19^2. \] Step-by-step solution: 1. Since the number ends in 2, it is even. Dividing by 2: \[ \frac{59,\!930,\!332}{2} = 29,\!965,\!166. \] 2. The quotient ends with 6, so it is even. Dividing by 2 again: \[ \frac{29,\!965,\!166}{2} = 14,\!982,\!583. \] At this point, we have: \[ 59,\!930,\!332 = 2^2 \cdot 14,\!982,\!583. \] 3. Testing divisibility of \(14,\!982,\!583\) by 7, note that: \[ 7 \times 2,\!140,\!369 = 14,\!982,\!583. \] Thus, \[ 14,\!982,\!583 = 7 \cdot 2,\!140,\!369. \] 4. Next, factor \(2,\!140,\!369\). Dividing by 7 again gives: \[ 2,\!140,\!369 = 7 \times 305,\!767. \] So far, we have: \[ 59,\!930,\!332 = 2^2 \cdot 7^2 \cdot 305,\!767. \] 5. Dividing \(305,\!767\) by 7: \[ 305,\!767 = 7 \times 43,\!681. \] Now the factorization is: \[ 59,\!930,\!332 = 2^2 \cdot 7^3 \cdot 43,\!681. \] 6. Notice that \(43,\!681\) is a perfect square, since: \[ 209^2 = 43,\!681. \] And since \(209 = 11 \cdot 19\), we have: \[ 43,\!681 = 11^2 \cdot 19^2. \] 7. Combining all factors, the prime-power representation is: \[ 59,\!930,\!332 = 2^2 \cdot 7^3 \cdot 11^2 \cdot 19^2. \]