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.)

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To determine the prime-power representation of each of the given integers, we need to express each integer as a product of prime numbers raised to certain powers.

(a) The prime-power representation of $$ 59,930,332 $$ is given as $$ 2^{2} \cdot 7^{3} \cdot 11^{2} \cdot 19^{2} $$.

(b) The prime-power representation of 48,400 is $$ 2^{4} \cdot 5^{2} \cdot 11^{2} $$.

(c) The prime-power representation of $$ 1,882,384 $$ is $$ 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2} $$.

(d) To determine which numbers in parts (a), (b), and (c) are squares, we need to check if the exponents of the prime factors are even.

(a) $$ 59,930,332 $$ is not a square because the exponents of the prime factors are not even.

(b) 48,400 is a square because the exponents of the prime factors are even.

(c) $$ 1,882,384 $$ is not a square because the exponents of the prime factors are not even.

(e) The prime-power representation of a square will have even exponents for all prime factors.

(f) To determine if $$ 27,543,608 $$ is the cube of a natural number, we need to check if the exponents of the prime factors are multiples of 3.

Let's calculate the prime-power representation of $$ 27,543,608 $$ to determine if it is the cube of a natural number.
Numbers by following steps:
- step0: Solution:
  $$27543608$$
The prime-power representation of $$ 27,543,608 $$ is $$ 2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 11^{3} $$.

Since the exponents of the prime factors are multiples of 3, $$ 27,543,608 $$ is the cube of a natural number.
(a) $$ 59,930,332 = 2^{2} \cdot 7^{3} \cdot 11^{2} \cdot 19^{2} $$ (b) $$ 48,400 = 2^{4} \cdot 5^{2} \cdot 11^{2} $$ (c) $$ 1,882,384 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2} $$ (d) Only 48,400 is a square. (e) A square has even exponents for all prime factors. (f) $$ 27,543,608 = 2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 11^{3} $$, which is the cube of a natural number.

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