First, when we round to the nearest thousand we get 5, 679, 000 since we round up when the next digit is greater than 5.

Then, to put it in scientific notation, we arrange the digits so that a decimal point creates a number between 1 and 10. We get 5.679.

Then, we want the exponent of the 10 to be the number of times the decimal needs to move to the right. This is 6 times.

Thus, we get our answer. $\displaystyle 5.679 \times 10^6$ 

In order to write a number in scientific notation, you must shift the number of decimal places to get a number in the ones place.

Since the original number is a decimal, the exponent will need to be negative. This eliminates three answer choices. 

In order to get $\displaystyle 0.000000537$ into scientific notation with '5' in the ones place, you must shift the decimal over seven places.

Therefore, the final answer in scientific notation is $\displaystyle 5.37 \times 10^{-7}$.

For this, you can handle the $\displaystyle 10$s separately from the coefficients.  The $\displaystyle 10$s easily cancel:

$\displaystyle \frac{1.57 * 10 ^3^3}{5*10^4^0} = \frac{1.57}{5*10^7}$

Now, handle the coefficients like they are a separate fraction:

$\displaystyle \frac{1.57}{5*10^7} = \frac{1.57}{5} * \frac{1}{10^7} = 0.314 * \frac{1}{10^7}$

Notice, though, that $\displaystyle \frac{1}{10^7} = 10^-^7$:

$\displaystyle 0.314 * \frac{1}{10^7} = 0.314 * 10^-^7$

Now, you need to get this into scientific notation for your answer.  It is easiest ot think of it like this:

$\displaystyle 0.314 * 10^-^7 = 3.14 * 10^-^1 * 10^-^7 = 3.14 * 10^-^8$

The easiest way to work with a large number like this is to combine the $\displaystyle 10$ factors and then the other coefficients. After that, you will have to convert the number into scientific notation to match the answer:

$\displaystyle (8.14 * 10^3^3) * (7.8 * 10^5^4) = 10^3^3 * 10^5^4 * 8.14*7.8$

Combine the powers of $\displaystyle 10$:

$\displaystyle 10^3^3 * 10^5^4 * 8.14*7.8 = 10^8^7 * 8.14*7.8$

Next, combine the other coefficients:

$\displaystyle 10^8^7 * 8.14*7.8 = 63.492 * 10^8^7$

Now, rewrite the leading number like this:

$\displaystyle 6.3492 * 10 * 10^8^7$

Thus, you get:

$\displaystyle 6.3492 * 10^8^8$

The easiest way to solve a question like this is to convert the answers all into easily comparable numbers. Remember that you move the decimal place to the left for negative powers of $\displaystyle 10$ and to the right for positive powers. Thus, for your number, you have:

$\displaystyle 114.1481 * 10^-^2 = 1.141481$

Notice that the negative power has no influence on the sign of the number. This allows you to eliminate the negative options. The correct answer is:

$\displaystyle 0.0001141481*10^4$

When you move the decimal $\displaystyle 4$ places to the right, you also get:

$\displaystyle 1.141481$