Question

Multiply and express the product in scientific notation:
$3,800,000\times 27,000$

• A. $10.26\times {10}^{10}$
• B. $10.26\times {10}^{11}$
• C. $1.026\times {10}^{10}$
• D. $1.026\times {10}^{11}$

Answer 1

The correct option is D $1.026\times {10}^{11}$

Let's start by writing the numbers in their simplest form.

$\bullet 3,800,000$
$=3,800,000.0$
Now, move the decimal point $6$ places to the left. Hence, the power to the base $10$ should be $6$.
$=3.8000000\times {10}^6$
$\Rightarrow \underset{–}{3,800,000=3.8\times {10}^6}$

$\bullet 27,000$
$=27,000.0$
Now, move the decimal point $4$ places to the left. Hence, the power to the base $10$ should be $4$.
$=2.70000\times {10}^4$
$\Rightarrow \underset{–}{27,000=2.7\times {10}^4}$

$\bullet 3,800,000\times 27,000$
$=(3.8\times {10}^6)\times (2.7\times {10}^4)$
$=(3.8\times 2.7)\times ({10}^6\times {10}^4)$

Applying $a^m\times a^n=a^{m+n}$ on $({10}^6\times {10}^4):$

$=\left(10.26\right)\times \left({10}^{6+4}\right)$

Now, for $10.26$, move the decimal point $1$ place to the left. Hence, the power to the base $10$ should be $1$.

$=1.026\times {10}^1\times {10}^{10}$
$=1.026\times {10}^{1+10}$ $\because (a^m\times a^n=a^{m+n})$

$=1.026\times {10}^{11}$

Hence, the value of the product $3,800,000\times 27,000$ is $\underset{–}{1.026\times {10}^{11}}.$

$\therefore$ Option (d.) is the correct one.