# Addition and Subtraction with Scientific Notation

Adding and subtracting with scientific notation may require more care, because the rule for adding and subtracting exponential expressions is that the expressions must have**like terms**. Remember that to be **like terms**, two expressions must have exactly the same base numbers to exactly the same powers. Thinking about decimal arithmetic, the requirement that we have the same powers makes sense, because that guarantees that all of the place values are lined up properly.

Example: (4.5 × 10^{4}) + (1.75 × 10^{4}) can be completed using the distributive property of multiplication over addition, i.e., factor out the common factor 10^{4}.

We run into trouble, though, with problems like (7.5 × 10^{3}) + (5.25 × 10^{5}) because the powers of 10 differ, so we need to modify the problem before we factor. We work around this by using our exponent property *b*^{m} · *b*^{n} = *b* ^{(m+n)} to rewrite the 10^{5} as 10^{2} · 10^{3} and then grouping the 10^{2} with the 5.25.

We see that this solution is not in standard scientific notation form because the decimal part has more than one digit in front of the decimal point. So we have one more step to finish the problem. We need to rewrite 532.5 as 5.325 × 10^{2} and then simplify the powers of ten.

Continue from above:

Subtraction can be done the same way as addition, by getting the powers of ten to match; factoring out the power of ten that is the same, and subtracting the decimal values that come together when the power of ten is factored out. Then we simplify if the answer is not in normalized form.

## Self-Check Problem

Approximately, how much further from the sun is Saturn than Earth. Earth is approximately 9.3 × 10^{7} miles from the sun and Saturn is approximately 8.87 × 10^{8} miles from the sun.

Solution

### Joke or Quote

**Several scientists were all posed the following question: "What is 2 times 2?"**

**The engineer whips out his slide rule (so it's old) and shuffles it back and forth, and finally announces "3.99".**

**The physicist consults his technical references, sets up the problem on his computer, and announces "it lies between 3.98 and 4.02".**

**The mathematician cogitates for a while, then announces: "I don't know what the answer is, but I can tell you, an answer exists!".**

**Philosopher smiles: "But what do you mean by 2 times 2?"**

**Logician replies: "Please define 2 times 2 more precisely." **

**The sociologist: "I don't know, but is was nice talking about it".**

**Behavioral Ecologist: "A polygamous mating system".**

**Medical Student: "Four". All others looking astonished ask "How did you know??" Medical Student: "I memorized it."**

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