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

Question

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

UpStudy AI · Accepted Answer

$$
59,\!930,\!332 = 2^2 \cdot 7^3 \cdot 11^2 \cdot 19^2.
$$

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:
   $$
   59,\!930,\!332 = 2^2 \cdot 7^3 \cdot 11^2 \cdot 19^2.
   $$
$$59,\!930,\!332 = 2^2 \cdot 7^3 \cdot 11^2 \cdot 19^2$$.

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