ADDING AND SUBTRACTING WITH SCIENTIFIC NOTATION

How to add and subtract numbers in scientific notation with same exponent ?

Let us consider the addition of two numbers in scientific notation. 

(9.35 x 10³) + (8.34 x 10³)

Factor 10³ out. 

Then,

=  (8.35 + 9.34) x 10³

=  17.69 x 10³

Write the above number in scientific notation. 

=  1.769 x 10⁴

Therefore, 

(9.35 x 10³) + (8.34 x 10³)  =  17.69 x 10³

Note : 

The same process has to be followed for subtracting numbers in scientific notation.   

How to add and subtract numbers in scientific notation with different exponents ?

Step 1 : 

Adjust the exponents of 10 in the given numbers such that they have the same exponent. 

(Tip : Always it is easier to adjust the smaller exponent to equal the larger exponent). 

Step 2 : 

In step 2, you will have the same exponent for 10 in all the numbers. For example, 10ⁿ

So, factor 10ⁿ out from all the numbers. 

Step 3 : 

Now, add or subtract the numbers and write the final answer in scientific notation. 

Solved Examples

Example 1 :

Evaluate :

(1.328 x 10⁷) + (2.034 x 10⁵)

Give your answer in scientific notation. 

Solution :

(1.328 x 10⁷) + (2.034 x 10⁵)

In the given numbers, we don't have the same exponent for 10.

Adjust the exponents of 10 in the given numbers such that they have the same exponent.

It is easier to adjust the smaller exponent to equal the larger exponent. 

Then, 

=  (1.328 x 10⁷) + (0.02034 x 10⁷)

In the above numbers, we have the same exponent for 10.

So, factor 10⁷ out from the given numbers. 

=  (1.328 + 0.02034) x 10⁷

  =  1.34834 x 10⁷

The above number is in scientific notation.

Therefore, 

(1.328 x 10⁷) + (2.034 x 10⁵)  =  1.34834 x 10⁷

Example 2 :

Evaluate :

(3.2 x 10^-3) - (8.02 x 10^-5)

Give your answer in scientific notation. 

Solution :

(3.2 x 10^-3) - (8.02 x 10^-5)

In the given numbers, we don't have the same exponent for 10.

Adjust the exponents of 10 in the given numbers such that they have the same exponent.

Then, 

=  (3.2 x 10^-3) - (0.0802 x 10² x 10^-5)

=  (3.2 x 10^-3) - (0.0802 x 10^2-5)

=  (3.2 x 10^-3) - (0.0802 x 10^-3)

In the above numbers, we have the same exponent for 10.

So, factor 10^-3 out from the given numbers. 

=  (3.2 - 0.0802) x 10^-3

=  3.1198 x 10^-3

Write the above number in scientific notation. 

=  3.1198 x 10^-3

Therefore, 

(3.2 x 10^-3) - (8.02 x 10^-5)  =  3.1198 x 10^-3

Example 3 : 

Simplify the expression given below. 

(0.723 x 10⁸) + (338.2 x 10⁵) - (6.1 x 10⁷)

Solution :

Method 1 :

Step 1 : 

In the given numbers, the highest power of 10 is 8. 

So, write each number with 10 power 8.

0.723 x 10⁸  =  0.723 x 10⁸

338.2 x 10⁵  =  0.3382 x 10⁸

6.1 x 10⁷  =  0.61 x 10⁸

Step 2 : 

Simplify the multipliers.

0.723 + 0.3382 - 0.61  =  0.4512

Step 3 : 

Write the final answer in scientific notation :

0.4512 x 10⁸  =  4.512 x 10⁷

Method 2 :

Step 1 : 

First, write each number in standard notation.

0.723 x 10⁸  =  72,300,000

338.2 x 10⁵  =  33,820,000

6.1 x 10⁷  =  61,000,000

Step 2 : 

Simplify the numbers in standard notation.

72,300,000 + 33,820,000 - 61,000,000  =  45,120,000

Step 3 : 

Write the final answer in scientific notation :

45,120,000  =  4.512 x 10⁷

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