When rolling a standard number cube: P(3 and then even)?

If you're solving this by hand using long division, the first step is to eliminate the decimal fraction in the divisor. You do this by moving the decimal point right 6 places in both divisor and dividend, making the problem be ...

... 27 ÷ 9 = 3

___

If you're solving this using a calculator (this is College math), you simply enter the numbers given and execute the appropriate operation:

... .000027 ÷ .000009 = . . . . . the result will be 3

<em>Scientific notation</em>

When a number is written in scientifc notation such that the mantissa has one digit to the left of the decimal point, the exponential term gives you the place value multiplier of that digit. For example, the 3 in 3.12×10⁴ has a place-value multiplier of 10⁴ = 10,000. This means the number written in standard form is ...

... 31,200 . . . . . with the 3 in the ten-thousands place

___

In this problem, the number 0.000027 has the 2 in the hundred-thousandths place, so its multiplier is 1/100,000 = 1/10⁵ = 10⁻⁵. In scientific notation, this number is written ...

... 2.7×10⁻⁵

Likewise, the number 0.000009 has the 9 in the millionths place, so its multiplier is 1/1,000,000 = 1/10⁶ = 10⁻⁶. In scientific notation, this number is written ...

... 9×10⁻⁶

Then the division of these numbers can be written as

$\dfrac{2.7\cdot 10^{-5}}{9\cdot 10^{-6}}$

<em>Another way to look at it</em>

The rules of exponents work the same way they would if the exponential terms were a⁴ and a³:

... (2.7a⁴)/(9a³) = (2.7/9)a⁽⁴⁻³⁾ = 0.3a

With negative exponents, we still subtract the denominator exponent from the numerator exponent:

... (2.7a⁻⁵)/(9a⁻⁶) = 0.3a⁽⁻⁵⁻⁽⁻⁶⁾⁾ = 0.3a¹ = 0.3a

The problem we have above has the numbers 2.7×10⁻⁵ instead of 2.7a⁻⁵. If we let a=10, the problems become identical. The solution is identical, too.

<em>Our problem</em>

$\dfrac{2.7\cdot 10^{-5}}{9\cdot 10^{-6}}=\dfrac{2.7}{9}\cdot 10^{(-5-(-6))}=0.3\cdot 10^{1}\\\\=0.3\cdot 10=3$

Summary

<em>How to divide numbers in scientific notation</em>

The mantissas are divided in the usual way. The exponential terms are divided according to the rules of exponents: the exponent of the denominator is subtracted from the exponent of the numerator to give the exponent of the result. Then, some adjustment may need to be made to the answer to put it into an appropriate form.

<em>Final answer in simplest form</em>

... 2.7×10⁻⁵/(9×10⁻⁶) = 0.3×10¹ = 3

_____

<em>Comment on the attachment</em>

Your calculator can work this problem in whatever fashion you choose. The numbers can be entered as decimal fractions or in scientific notation. The results can be displayed in standard form or in scientific notation.