Question

How many zeros are at the end of457 · 885?Explain how you can answer this question without actually computing the number. (Hint:10 = 2 · 5.)When this number is written in ordinary decimal form, each 0 at its end comes from a factor of , or one factor of 2 and one factor of .Since there are factors of 2 and factors of 5, there are exactly factors of 10 in the number. This implies that the number ends with zeroes.

Answer 1

The question is not well formatted. However, I have taken some educated guesses as to what it should be.

Since it requires the number of zeros in a product, one of them should be a multiple of 5 and the other an even number. Both numbers appear as odd numbers, so I believe the first number is $45^7$ while the other is either $8^{85}$ or $88^5$.

Answer:

The product has 7 zeros

Step-by-step explanation:

The number of zeros is determined by the number of 10 in the factors. Now 10 = 2× 5. Hence, it is determined by the least number of factors of both 2 and 5.

45 = 5 × 9

$45^7 = (5\times9)^7 = 5^7\times9^7$

Hence $45^7$ has 7 factors of 5.

$88^5 = (2^3\times11)^5 = 2^{15}\times11^{15}$

Here, this has 15 factors of 2.

If we take the second number to be $8^{85}$, then

$8^{85} =(2^3)^{85} = 2^{255}$.

The number of the factors of 10 in the product is the minimum of the number of factors of 2 and 5. Here, it is 7.

To fill in the blanks in the question:

When this number is written in ordinary decimal form, each 0 at its end comes from a factor of 10, or one factor of 2 and one factor of 5. Since there are 15 or 255 (make a choice, depending on the correct question) factors of 2 and 7 factors of 5, there are exactly 7 factors of 10 in the number. This implies that the number ends with 7 (seven) zeroes.

Answer 2

The right format of the number is (45^8)(88^5).

Answer:

There are 8 zeros

Step-by-step explanation:

Using the unique factorization of integers theorem, we can break any integer down into the product of prime integers.

So breaking it down we have;

(45^8) = (3 x 3 x 5)^(8)

(88^5) = (2 x 2 x 2 x 11)^(5)

Now, if we put it back together as separate factors, we'll get;

(3^(16)) x (5^(8) ) x (2^(15)) x (11^(5))

Now let's find the number of zeroes by figuring out how many factors of 10 (which equals 2 x 5) we can make. Thus, we can make 8 factors of 10 so it looks like;

(3^(16)) x (2^(7)) x (11^(5)) x (10^(8))

Thus, we can see that there will be 8 zeros as the end is (10^(8))