We want to add two numbers that are written in scientific notation.

(8.2 \times 1 0^{2}) + (3.41 \times 1 0^{- 1})

Notice that the exponent of

10

is different for both numbers. Therefore, to add the numbers we will need to first rewrite the first number as a multiple of

1 0^{- 1} .

To do so, we will use the Product of Powers Property.

(8.2 \times 1 0^{2}) + (3.41 \times 1 0^{- 1})

Write as a sum

(8.2 \times 1 0^{- 1 + 3}) + (3.41 \times 1 0^{- 1})

$a^{m + n} = a^{m} \cdot a^{n}$

(8.2 \times 1 0^{- 1} \cdot 1 0^{3}) + (3.41 \times 1 0^{- 1})

Commutative Property of Multiplication

(8.2 \times 1 0^{3} \cdot 1 0^{- 1}) + (3.41 \times 1 0^{- 1})

Calculate power

(8.2 \times 1000 \cdot 1 0^{- 1}) + (3.41 \times 1 0^{- 1})

(8200 \times 1 0^{- 1}) + (3.41 \times 1 0^{- 1})

Now,

10

has the same exponent in both numbers. To add the expressions, we can factor out

1 0^{- 1} .

(8200 \times 1 0^{- 1}) + (3.41 \times 1 0^{- 1})

Factor out $1 0^{- 1}$

(8200 + 3.41) \times 1 0^{- 1}

Add terms

8203.41 \times 1 0^{- 1}

Recall that the first factor of a number written in scientific notation must be greater than or equal to

1

and less than

10 .

Therefore, we need to rewrite

8203.41

as the product of

8.20341

and

1000 .

Then, we can simplify.

8203.41 \times 1 0^{- 1}

Rewrite $8203.41$ as $8.20341 \cdot 1000$

8.20341 \cdot 1000 \times 1 0^{- 1}

Write as a power

8.20341 \cdot 1 0^{3} \times 1 0^{- 1}

$a^{m} \cdot a^{n} = a^{m + n}$

8.20341 \times 1 0^{3 + (- 1)}

$a + (- b) = a - b$

8.20341 \times 1 0^{2}

Exercise