Question

It is often necessary to do calculations using scientific notation when working chemistry problems. For practice, perform each of the following calculations.

1. 7.79×10-4 + 7.63×105/ 7.54×10-4=

2. 1.50×10-5/ (1.00×10-8)(9.82×104)=

please help!

Answer 1

Step-by-step explanation:

Scientific notation uses "powers of ten" to write very large or very small numbers in a shorter space.

When you see a "x 10" you know you are about to run into scientific notation.

* When the exponent of the 10 is positive, then you will multiply by 10 exactly that number of times. (e.g.,
`10^4 = * 10 * 10 * 10 * 10`)

* When the exponent of the 10 is negative, then you will divide by 10 exactly that number of times. (e.g.,
`10^(-4) = / 10 / 10 / 10 / 10`).

The Hard Way (Ignoring Significant Figures)

In the first problem, we have

`7.79 * 10^(-4) + 7.63 * 10^5 / 7.54 * 10^(-4)`

You can do this easily with a scientific calculator. I recommend the one from Desmos.

However, here's what it looks like by hand (so you can see the calculations happening):

Step 1: Convert the powers of 10.

`7.79 * 10^(-4) \rightarrow (7.79 / 10 / 10 / 10 / 10) \rightarrow 0.000779`

`7.63 * 10^5 \rightarrow 7.63 * 10 * 10 * 10 * 10 * 10 \rightarrow 763000`

`7.54 * 10^(-4) \rightarrow (7.54 / 10 / 10 / 10 / 10) \rightarrow 0.000754`

Step 2: Do math! :)

`7.79 * 10^(-4) + 7.63 * 10^5 / 7.54 * 10^(-4) = 0.000779 + 763000 / 0.000754`

`= 0.000779 + 1011936340 = 1011936340.000779`

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Now, if you do this the easy way on a calculator, you will get the answer:

`1.01193634 * 10^9`

which is basically the same as what we got, just written in scientific notation and rounded.

The Real Way (Using Significant Figures)

With significant figures, we have two primary rules:

1) Multiplying or dividing two numbers keeps the smallest number of significant figures.

2) Adding or subtracting two numbers keeps the smallest number of decimals.

This means that you will usually be rounding your answers.

So, on Step 2 above, using significant figures we would have seen:

`0.000779 + 1010000000`

since the two numbers we divided only had three significant digits. Then, our final answer would be:

`0.000779 + 1010000000 = 1010000000 = 1.01 * 10^9`

And, no, that isn't a mistake! Since we are adding two numbers, we keep the smallest number of decimals, the second giant number has no decimals at all, so they just disappear!

Oh yeah, there was a second question...

`1.50 * 10^(-5) / \big( (1.00 * 10^-8)(9.82 * 10^4) \big) = 1.50 * 10^(-5) / 9.82 * 10^(-4) = 0.0153`

since only three significant digits are allowed.