Table of Contents

Last modified on September 23rd, 2022

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.

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

- Check if the exponents of the numbers are equal or not.
- If they are equal, factor out the numbers using the distributive property of multiplication (a
^{m}x a^{n}= a^{m}^{+ n}). - 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 10 ^{3}) + (1.89 x 10^{3})**

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

(4.7 x 10^{3}) + (1.89 x 10^{3})

(4.7 + 1.89) × 10^{3}

- Adding the coefficients and multiplying by the power of 10, we get = 6.59 × 10
^{3}

Thus, the answer is 6.59 × 10^{3}

**With Unequal Exponents**

**Example – 2**

**Add: (8.8 x 10 ^{3}) + (6.25 x 10^{5})**

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 a^{n} = a^{m} ^{+ n}, rewriting the larger exponent 10^{5} into (10^{2} × 10^{3})

= (8.8 x 10^{3}) + (6.25 x 10^{5})

= (8.8 x 10^{3}) + (6.25 x 10^{2} × 10^{3})

= (8.8 x 10^{3}) + [(6.25 x 10^{2}) × 10^{3}]

= (8.8 x 10^{3}) + (625 × 10^{3})

= (8.8 + 625) × 10^{3}

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

= 633.8 × 10^{3}

Thus, the answer is 633.8 × 10^{3}

**(3.769 x 10 ^{5}) + (4.21 x 10^{5})**

Solution:

These two numbers have similar exponents

Using the distributive property of multiplication; the numbers are factored out as shown below:

(3.769 x 10^{5}) + (4.21 x 10^{5})

(3.769 + 4.21) × 10^{5}

Adding the coefficients and multiplying by the power of 10, we get = 7.979 × 10^{5}

Thus, the answer is 7.979 × 10^{5}

**Add: (5.5 x 10 ^{2}) + (1.25 x 10^{4})**

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: a^{m} x a^{n} = a^{m} ^{+ n}, rewriting the larger exponent 10^{5} into (10^{2} × 10^{3})

= (5.5 x 10^{2}) + (1.25 x 10^{4})

= (5.5 x 10^{2}) + (6.25 x 10^{2} × 10^{3})

= (8.8 x 10^{3}) + [(6.25 x 10^{2}) × 10^{3}]

= (8.8 x 10^{3}) + (625 × 10^{3})

= (8.8 + 625) × 10^{3}

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

= 633.8 × 10^{3}

Thus, the answer is 633.8 × 10^{3}

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})**

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

= (4.4 x 10^{2}) – (2.89 x 10^{2})

= (4.4 – 2.89) × 10^{2}

- Adding the coefficients and multiplying by the power of 10, we get = 1.51 × 10
^{2}

Thus, the answer is 1.51 × 10^{2}

**With Unequal Exponents**

**Example – 2**

**Subtract: (7.35 x 10 ^{7}) – (5.8 x 10^{4})**

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 a^{n} = a^{m} ^{+ n}, rewriting the larger exponent 10^{7} into (10^{3} × 10^{4})

= (7.35 x 10^{7}) – (5.8 x 10^{4})

= (7.35 x 10^{3} × 10^{4}) – (5.8 x 10^{4})

= [(7.35 x 10^{3}) × 10^{4}] – (5.8 x 10^{4})

= (7350 × 10^{4}) – (5.8 x 10^{4})

= (7350 – 5.8) × 10^{4}

Adding the coefficients and multiplying by the power of 10, we get = 7,344.2 × 10^{4}

Thus, the answer is 7,344.2 × 10^{4}

**Subtract: (6 x 10 ^{5}) – (4.49 x 10^{5})**

Solution:

These two numbers have similar exponents

Using the distributive property of multiplication; the numbers are factored out as shown below:

= (6 x 10^{5}) – (4.49 x 10^{5})

= (6 – 4.49) × 10^{5}

Adding the coefficients and multiplying by the power of 10, we get = 1.51 × 10^{5}

Thus, the answer is 1.51 × 10^{5}

**Subtract: (3.25 x 10 ^{5}) – (9.8 x 10^{2})**

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: a^{m} x a^{n} = a^{m} ^{+ n}, rewriting the larger exponent 10^{7} into (10^{3} × 10^{4})

= (3.25 x 10^{5}) – (9.8 x 10^{2})

= (3.25 x 10^{3} × 10^{2}) – (9.8 x 10^{2})

= [(3.25 x 10^{3}) × 10^{2}] – (9.8 x 10^{2})

= (3250 × 10^{2}) – (9.8 x 10^{4})

= (3250 – 9.8) × 10^{2}

Adding the coefficients and multiplying by the power of 10, we get = 3,240.2 × 10^{2}

Thus, the answer is 3,240.2 × 10^{2}