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 (a^m x aⁿ = a^{m + 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) × 10³

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

Thus, the answer is 6.59 × 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:  a^m x aⁿ = a^{m + n}, rewriting the larger exponent 10⁵ into (10² × 10³)

= (8.8 x 10³) + (6.25 x 10⁵)

= (8.8 x 10³) + (6.25 x 10² × 10³)

= (8.8 x 10³) + [(6.25 x 10²) × 10³]

= (8.8 x 10³) + (625 × 10³)

= (8.8 + 625) × 10³

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

 = 633.8 × 10³

Thus, the answer is 633.8 × 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) × 10²

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

Thus, the answer is 1.51 × 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:  a^m x aⁿ = a^{m + n}, rewriting the larger exponent 10⁷ into (10³ × 10⁴)

= (7.35 x 10⁷) – (5.8 x 10⁴)

= (7.35 x 10³ × 10⁴) – (5.8 x 10⁴)

= [(7.35 x 10³) × 10⁴] – (5.8 x 10⁴)

= (7350 × 10⁴) – (5.8 x 10⁴)

= (7350 – 5.8) × 10⁴

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

Thus, the answer is 7,344.2 × 10⁴