# Adding and Subtracting Numbers in Scientific Notation

Adding or subtracting numbers in scientific notation requires the numbers to have similar bases and exponents. This is necessary to ensure that the corresponding integers in their coefficients have the same place value.

## Adding in Scientific Notation with Examples

A few steps we should follow while adding numbers in scientific notation are:

1. Check if the exponents of the numbers are equal or not.
2. If they are equal, factor out the numbers using the distributive property of multiplication (am x an = am + n).
3. Add the coefficients. If unequal, the exponents should be made equal by moving the decimal point. The easiest way to make the decimals equal is to make the smaller exponent equal to the larger exponent by shifting the decimal to the left.

Let us understand the concept with a few examples.

With Equal Exponents

Example – 1

Add: (4.7 x 103) + (1.89 x 103)

1. These two numbers have similar exponents
2. Using the distributive property of multiplication; the numbers are factored out as shown below:

(4.7 x 103) + (1.89 x 103)

(4.7 + 1.89) Ã— 103

• Adding the coefficients and multiplying by the power of 10, we get = 6.59 Ã— 103

Thus, the answer is 6.59 Ã— 103

With Unequal Exponents

Example – 2

Add:  (8.8 x 103) + (6.25 x 105)

Here, the powers of the numbers are different. Thus we need to manipulate the power with a larger exponent.

Using the property of exponents:  am x an = am + n, rewriting the larger exponent 105 into (102 Ã— 103)

= (8.8 x 103) + (6.25 x 105)

= (8.8 x 103) + (6.25 x 102 Ã— 103)

= (8.8 x 103) + [(6.25 x 102) Ã— 103]

= (8.8 x 103) + (625 Ã— 103)

= (8.8 + 625) Ã— 103

Adding the coefficients and multiplying by the power of 10, we get

= 633.8 Ã— 103

Thus, the answer is 633.8 Ã— 103

(3.769 x 105) + (4.21 x 105)

Solution:

These two numbers have similar exponents
Using the distributive property of multiplication; the numbers are factored out as shown below:
(3.769 x 105) + (4.21 x 105)
(3.769 + 4.21) Ã— 105
Adding the coefficients and multiplying by the power of 10, we get = 7.979 Ã— 105
Thus, the answer is 7.979 Ã— 105

Add:Â  (5.5 x 102) + (1.25 x 104)

Solution:

Here, the powers of the numbers are different. Thus we need to manipulate the power with a larger exponent.
Using the property of exponents:Â  amÂ x anÂ = am + n, rewriting the larger exponent 105 into (102 Ã— 103)
= (5.5 x 102) + (1.25 x 104)
= (5.5 x 102) + (6.25 x 102 Ã— 103)
= (8.8 x 103) + [(6.25 x 102) Ã— 103]
= (8.8 x 103) + (625 Ã— 103)
= (8.8 + 625) Ã— 103
Adding the coefficients and multiplying by the power of 10 we get
Â = 633.8 Ã— 103
Thus, the answer is 633.8 Ã— 103

## Subtracting in Scientific Notation with Examples

The basic steps to subtract two or more numbers in scientific notation are the same as that of addition, except that addition is replaced by subtraction.

Let us consider a few examples.

With Equal Exponents

Example – 1

Subtract: (4 x 10-2) – (2.89 x 10-2)

1. These two numbers have similar exponents
2. Using the distributive property of multiplication; the numbers are factored out as shown below:

= (4.4 x 102) – (2.89 x 102)

= (4.4 – 2.89) Ã— 102

• Adding the coefficients and multiplying by the power of 10, we get = 1.51 Ã— 102

Thus, the answer is 1.51 Ã— 102

With Unequal Exponents

Example – 2

Subtract:  (7.35 x 107) – (5.8 x 104)

Here, the powers of the numbers are different. Thus we need to manipulate the power with a larger exponent.

Using the property of exponents:  am x an = am + n, rewriting the larger exponent 107 into (103 Ã— 104)

= (7.35 x 107) – (5.8 x 104)

= (7.35 x 103 Ã— 104) – (5.8 x 104)

= [(7.35 x 103) Ã— 104] – (5.8 x 104)

= (7350 Ã— 104) – (5.8 x 104)

= (7350 – 5.8) Ã— 104

Adding the coefficients and multiplying by the power of 10, we get = 7,344.2 Ã— 104

Thus, the answer is 7,344.2 Ã— 104

Subtract: (6 x 105) – (4.49 x 105)

Solution:

These two numbers have similar exponents
Using the distributive property of multiplication; the numbers are factored out as shown below:
= (6 x 105) – (4.49 x 105)
= (6 – 4.49) Ã— 105
Adding the coefficients and multiplying by the power of 10, we get = 1.51 Ã— 105
Thus, the answer is 1.51 Ã— 105

Subtract: (3.25 x 105) – (9.8 x 102)

Solution:

Here, the powers of the numbers are different. Thus we need to manipulate the power with a larger exponent.
Using the property of exponents:Â  amÂ x anÂ = am + n, rewriting the larger exponent 107 into (103 Ã— 104)
= (3.25 x 105) – (9.8 x 102)
= (3.25 x 103 Ã— 102) – (9.8 x 102)
= [(3.25 x 103) Ã— 102] – (9.8 x 102)
= (3250 Ã— 102) – (9.8 x 104)
= (3250 – 9.8) Ã— 102
Adding the coefficients and multiplying by the power of 10, we get = 3,240.2 Ã— 102
Thus, the answer is 3,240.2 Ã— 102