Which expression is equivalent to [tex]\left(2 \cdot 10^6\right) \div \left(3 \cdot 10^{-2}\right) \cdot \left(3 \cdot 10^2\right)[/tex]?

A. [tex]2 \cdot 10^{-4}[/tex]
B. [tex]2 \cdot 10^4[/tex]
C. [tex]1.8 \cdot 10^6[/tex]
D. [tex]2 \cdot 10^{10}[/tex]



Answer :

Sure, let's break down the given expression step-by-step to find its equivalent form.

The expression provided is:

[tex]\[ \left(2 \cdot 10^6\right) \div \left(3 \cdot 10^{-2}\right) \cdot \left(3 \cdot 10^2\right) \][/tex]

First, let's start with the division part:

[tex]\[ \left(2 \cdot 10^6\right) \div \left(3 \cdot 10^{-2}\right) \][/tex]

We can simplify this as follows:

[tex]\[ \frac{2 \cdot 10^6}{3 \cdot 10^{-2}} \][/tex]

When dividing numbers in scientific notation, we can divide the coefficients and subtract the exponents in the denominator from the numerator. So,

[tex]\[ \frac{2}{3} \cdot 10^{6 - (-2)} = \frac{2}{3} \cdot 10^{6 + 2} = \frac{2}{3} \cdot 10^8 \][/tex]

Next, let's handle the multiplication part of the original expression:

[tex]\[ \left(\frac{2}{3} \cdot 10^8 \right) \cdot \left(3 \cdot 10^2\right) \][/tex]

Again, multiply the coefficients and add the exponents:

[tex]\[ \left(\frac{2}{3} \cdot 3\right) \cdot 10^{8 + 2} = 2 \cdot 10^{10} \][/tex]

Thus, the simplified form of the given expression is:

[tex]\[ 2 \cdot 10^{10} \][/tex]

Finally, looking at the options provided:

(A) [tex]\(2 \cdot 10^{-4}\)[/tex]
(B) [tex]\(2 \cdot 10^4\)[/tex]
(C) [tex]\(1.8 \cdot 10^6\)[/tex]
(D) [tex]\(2 \cdot 10^{10}\)[/tex]

The correct answer is:
[tex]\[ \boxed{D} \][/tex]