# How do you multiply expressions written in scientific notation?

Nov 29, 2014

Expressions can be easily multiplied when written in scientific notation by:
1. First, multiplying the numbers other than the powers of 10.
2. Second, multiplying the powers of 10
And then, writing them as a product.

Let us take the general case first.

Multiplying two numbers $x \cdot {10}^{m}$ and $y \cdot {10}^{n}$

First, multiplying the numbers other than the powers of 10, we get:
$x \cdot y = x y$

Second, multiplying the powers of 10 we get
${10}^{m} \cdot {10}^{n} = {10}^{m + n}$

And then writing them as a product, we get
$x y \cdot {10}^{m + n}$

Therefore, $\left(x \cdot {10}^{m}\right) \cdot \left(y \cdot {10}^{n}\right) = x y \cdot {10}^{m + n}$

Note: When the bases of 2 numbers are equal, their powers can be added up!
Examples:
1). ${2}^{a} \cdot {2}^{b} = {2}^{a + b}$
2) ${3}^{3} \cdot {3}^{7} = {3}^{3 + 7} = {3}^{10}$

Now, let's take some specific examples.

Q: Multiply $1.2 \cdot {10}^{3}$ and $2.3 \cdot {10}^{4}$

A:

$\left(1.2 \cdot {10}^{3}\right) \cdot \left(2.3 \cdot {10}^{4}\right)$
$= \left(1.2 \cdot 2.3\right) \cdot \left({10}^{3 + 4}\right)$
$= 2.76 \cdot {10}^{7}$

Q: Multiply $9.32 \cdot {10}^{21}$ and $8.21 \cdot {10}^{32}$

A:

$\left(9.32 \cdot {10}^{21}\right) \cdot \left(8.21 \cdot {10}^{32}\right)$
$= \left(9.32 \cdot 8.21\right) \cdot \left({10}^{21 + 32}\right)$
$= 76.5172 \cdot {10}^{53}$

Notice that this answer is not in the standard form. So, converting this into standard form, we get:

$= 7.65172 \cdot {10}^{54}$