Solve: (10^4 × 0.1^-3 × 10^-8) ÷ 1000 Using Scientific Notation

Begin by writing the problem and converting the decimal fraction in the problem to a simple fraction:

$\frac{10^4\cdot0.1^{-3}\cdot10^{-8}}{1000}=\frac{10^4\cdot(\frac{1}{10})^{-3}\cdot10^{-8}}{1000}=\text{?}$

Next

a. We'll use the law of exponents for negative exponents:

$a^{-n}=\frac{1}{a^n}$

b. Note that the number 1000 is a power of the number 10:

$1000=10^3$

Apply the law of exponents from 'a' and the information from 'b' to the problem:

$\frac{10^4\cdot(\frac{1}{10})^{-3}\cdot10^{-8}}{1000}=\frac{10^4\cdot(10^{-1})^{-3}\cdot10^{-8}}{10^3}$

We applied the law of exponents from 'a' to the term inside the parentheses of the middle term in the fraction's numerator. We applied the information from 'b' to the fraction's denominator,

Next, let's recall the law of exponents for power of a power:

$(a^m)^n=a^{m\cdot n}$

And we'll apply this law to the same term we dealt with until now in the expression that we obtained in the last step:

$\frac{10^4\cdot(10^{-1})^{-3}\cdot10^{-8}}{10^3}=\frac{10^4\cdot10^{(-1)\cdot(-3)}\cdot10^{-8}}{10^3}=\frac{10^4\cdot10^3\cdot10^{-8}}{10^3}$

We applied the above law of exponents to the middle term in the numerator carefully, since the term in parentheses has a negative exponent. Hence we used parentheses and then proceeded to simplify the resulting expression,

Note that we can reduce the middle term in the fraction's numerator with the fraction's denominator. This is possible due to the fact that a multiplication operation exists between all terms in the fraction's numerator. Let's proceed to reduce:

$\frac{10^4\cdot10^3\cdot10^{-8}}{10^3}=10^4\cdot10^{-8}$

Let's summarize the various steps to our solution so far:

$\frac{10^4\cdot(\frac{1}{10})^{-3}\cdot10^{-8}}{1000}=\frac{10^4\cdot(10^{-1})^{-3}\cdot10^{-8}}{10^3} = \frac{10^4\cdot10^3\cdot10^{-8}}{10^3} = 10^4\cdot10^{-8}$

Remember the law of exponents for multiplication of terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's apply this law to the expression that we obtained in the last step:

$10^4\cdot10^{-8}=10^{4+(-8)}10^{4-8}=10^{-4}$

Now let's once again apply the law of exponents for negative exponents mentioned in 'a' above:

$10^{-4}=\frac{1}{10^4}=\frac{1}{10000}=0.0001$

When in the third step we calculated the numerical result of raising 10 to the power of 4 in the fraction's denominator. In the next step we converted the simple fraction to a decimal fraction,

Let's summarize the various steps of our solution so far:

$\frac{10^4\cdot(\frac{1}{10})^{-3}\cdot10^{-8}}{1000} = \frac{10^4\cdot10^3\cdot10^{-8}}{10^3} = 10^{-4} =0.0001$

Therefore the correct answer is answer a.