# How do you simplify and write 0.0007 xx 190 in scientific notation?

May 23, 2017

See a solution process below:

#### Explanation:

First, write each term in scientific notation:

For $0.0007$ we need to move the decimal point 4 places to the right therefore the exponent of the 10s term will be negative:

$0.0007 = 7.0 \times {10}^{-} 4$

For $190$ we need to move the decimal point 2 places to the left therefore the exponent of the 10s term will be positive:

$190 = 1.9 \times {10}^{2}$

We can now rewrite this expression as:

$0.0007 \times 190 \implies \left(7.0 \times {10}^{-} 4\right) \left(1.9 \times {10}^{2}\right) \implies$

$\left(7.0 \times 1.9\right) \left({10}^{-} 4 \times {10}^{2}\right) \implies 13.3 \left({10}^{-} 4 \times {10}^{2}\right)$

We can now use this rule of exponents to combine the 10s terms:

${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$

$13.3 \left({10}^{\textcolor{red}{- 4}} \times {10}^{\textcolor{b l u e}{2}}\right) \implies 13.3 \times {10}^{\textcolor{red}{- 4} + \textcolor{b l u e}{2}} \implies 13.3 \times {10}^{-} 2$

To put this in true scientific notation we need to move the decimal point one place to the left so we need to add $1$ to the 10s exponent:

$13.3 \times {10}^{-} 2 \implies 1.33 \times {10}^{- 2 + 1} \implies 1.33 \times {10}^{-} 1$

May 23, 2017

A lot of detail provided to help with understanding.

"1.33xx10^(-1)

#### Explanation:

Note that ${10}^{0} = 1 \text{ and that } {10}^{1} = 10$

$0.0007 \to 0.0007 \times \frac{1}{10} ^ 0$

$0.0007 \to 0.007 \times \frac{1}{10} ^ 1$

$0.0007 \to 0.07 \times \frac{1}{10} ^ 2$

$0.0007 \to 0.7 \times \frac{1}{10} ^ 3$

$0.0007 \to 7.0 \times \frac{1}{10} ^ 4$
..............................................................................

$190 \to 190 \times {10}^{0}$

$190 \to 19.0 \times {10}^{1}$
...........................................................................

So we can write: $\text{ } 0.0007 \times 190$ as

$7 \times 19 \times \frac{1}{10} ^ 4 \times 10$

Not that $7 \times 19$ is the same as $\left(7 \times 20\right) - 7 = 140 - 7 = 133$ giving:

$133 \times \frac{1}{10} ^ 4 \times 10 \text{ } \to 133 \times \frac{1}{10} ^ 3$

but $133 \text{ is the same as } 1.33 \times {10}^{2}$ giving:

$1.33 \times {10}^{2} / {10}^{3}$

$1.33 \times \frac{1}{10} ^ \text{ "->" } 1.33 \times {10}^{- 1}$