# Scientific Notation

### Addition and Subtraction in Scientific Notation

A number written in scientific notation is written as the product of a number between 1 and 10 and a number that is a power of 10 . That is, it is written as a quantity whose coefficient is between 1 and 10 and whose base is 10 .

One of the properties of quantities with exponents is that numbers with exponents can be added and subtracted only when they have the same base and exponent. Since all numbers in scientific notation have the same base (10), we need only worry about the exponents. To be added or subtracted, two numbers in scientific notation must be manipulated so that their bases have the same exponent--this will ensure that corresponding digits in their coefficients have the same place value.

Multiplying a number by another number with the same base is equivalent to multiplying their coefficients and adding their exponents. Therefore, if we want to add two quantities written in scientific notation whose exponents do not match, we can simply write one of the powers of 10 as the product of two smaller powers of 10 , one of which agrees with the other term.

Alternately, if we want to preserve the exponent of the term with the larger power of 10 , we can simultaneously multiply and divide the other term by a power of 10 , applying the rule for multiplication of exponents in one case and dividing the coefficient in the other. It is this procedure that we outline below.

Once the numbers have the same base and exponents, we can add or subtract their coefficients.

Here are the steps to adding or subtracting numbers in scientific notation :

1. Determine the number by which to increase the smaller exponent by so it is equal to the larger exponent.
2. Increase the smaller exponent by this number and move the decimal point of the number with the smaller exponent to the left the same number of places. (i.e. divide by the appropriate power of 10 .)
3. Add or subtract the new coefficients.
4. If the answer is not in scientific notation (i.e. if the coefficient is not between 1 and 10 ) convert it to scientific notation.
The answer should include coefficient, base, and exponent.

Note: If the numbers start with the same exponents, their coefficients can be added, but be careful--the answer might need to be converted to scientific notation.

Example 1: 2.456×105 +6.0034×108 = ?

1. 8 - 5 = 3. The smaller exponent must be increased by 3.
2. 2.456×105 = 0.002456×108
3. 0.002456×108 +6.0034×108 = 6.005856×108
4. 6.005856×108 is in scientific notation.
Thus, 2.456×105 +6.0034×108 = 6.005856×108 .

Example 2: 5.10802×103 -6.1×10-2 = ?

1. 3 - (-2) = 5. The smaller exponent must be increased by 5.
2. 6.1×10-2 = 0.000061×103
3. 5.10802×103 -0.000061×103 = 5.107959×103
4. 5.107959×103 is in scientific notation.
Thus, 5.10802×103 -6.1×10-2 = 5.107959×103 .

Example 3: 3.5×1018 + 5.3×1012 = ?

1. 18 - 12 = 6. The smaller exponent must be increased by 6.
2. 5.3×1012 = 0.0000053×1018
3. 3.5×1018 + 0.0000053×1018 = 3.5000053×1018
4. 3.5000053×1018 is in scientific notation.
Thus, 3.5×1018 + 5.3×1012 = 3.5000053×1018 .

Example 4: 4.801×103 -2.2×107 = ?

1. 7 - 3 = 4. The smaller exponent must be increased by 4.
2. 4.801×103 = 0.0004801×107
3. 0.0004801×107 -2.2×107 = - 2.1995199×107
4. -2.1995199×107 is in scientific notation.
Thus, 4.801×103 -2.2×107 = - 2.1995199×107 .

Example 5. 1.4×10-5 -5.67×10-6 = ?

1. -5 - (-6) = 1. The smaller exponent must be increased by 1.
2. 5.67×10-6 = 0.567×10-5
3. 1.4×10-5 -0.567×10-5 = 0.833×10-5
4. 0.833×10-5 = 8.33×10-6 in scientific notation.
Thus, 1.4×10-5 -5.67×10-6 = 8.33×10-6 .

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