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

First let's write the problem and convert 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:

$1000=10^3$

Let's apply the law of exponents from 'a' and the understanding from 'b' to the problem:

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

When we applied the law of exponents from 'a' to the term inside the parentheses of the middle term in the fraction's numerator, and applied the understanding 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 we got in the last step:

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

When we applied the above law of exponents to the middle term in the numerator carefully, since the term in parentheses has a negative exponent, we used parentheses, then simplified the resulting expression,

Now note that we can reduce the middle term in the fraction's numerator with the fraction's denominator, this is possible because multiplication exists between all terms in the fraction's numerator, so let's reduce:

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

Let's summarize the solution steps so far, we got that:

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

Let's continue and recall 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 we got in the last step:

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

Now let's apply again 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, and in the next step we converted the simple fraction to a decimal fraction,

Let's summarize the solution steps so far, we got that:

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

Therefore the correct answer is answer a.