# How do you write the answer in scientific notation given (3.4times10^-1)(3.1times10^-2)?

Jan 9, 2018

$1.054 \cdot {10}^{-} 2$

#### Explanation:

Scientific notation is a notation which can be written as

$x \cdot {10}^{n}$

where $x \in \left[1 , 10\right)$ .

When multiplying, the first thing to do is to multiply the like terms.

$3.4 \cdot 3.1 = 10.54$
${10}^{-} 1 \cdot {10}^{-} 2 = {10}^{-} 3$

So the answer looks like it should be

$10.54 \cdot {10}^{-} 3$

but it isn't, because $x$ (as defined earlier), is not between 1 and 10. So we divide $x$ by 10 and add one to that power to get the final answer:

$1.054 \cdot {10}^{-} 2$

Jan 9, 2018

$1.054 \times {10}^{-} 2$

#### Explanation:

Consider the product: $\text{ } 3 {x}^{4} \times 5 {x}^{7}$

In algebra we multiply the numbers:

$\text{ } 3 \times 5 = 15$

Then add the indices of bases that are the same:

${x}^{4} \times {x}^{7} = {x}^{4 + 7} = {x}^{11}$

$3 {x}^{4} \times 5 {x}^{7} = 15 {x}^{11}$

Scientific notation works in exactly the same way:

$\left(3.4 \times {10}^{-} 1\right) \times \left(3.1 \times {10}^{-} 2\right)$

Multiply the numbers:

$3.4 \times 3.1 = 10.54$

Add the indices of bases that are the same:

${10}^{-} 1 \times {10}^{-} 2 = {10}^{- 1 - 2} = {10}^{- 3}$

$\left(3.4 \times {10}^{-} 1\right) \times \left(3.1 \times {10}^{-} 2\right) = 10.54 \times {10}^{-} 3$

However a value in scientific notation must only have one digit before the decimal point. it is written in the form:

$a \times {10}^{n} , \text{ }$ where $1 \le a < 10 \mathmr{and} n \in \mathbb{Z}$

$10.54 \times {10}^{-} 3 = 1.054 \times {10}^{-} 2$