**Adding and Subtracting with Scientific Notation : **

In this section, you will learn how to add and subtract numbers in scientific notation.

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

(9.35 x 10^{3}) + (8.34 x 10^{3})

Factor 10^{3} out.

Then,

= (8.35 + 9.34) x 10^{3}

= 17.69 x 10^{3}

Write the above number in scientific notation.

= 1.769 x 10^{4}

Therefore,

(9.35 x 10^{3}) + (8.34 x 10^{3}) = 17.69 x 10^{3}

**Note : **

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

**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, 10^{n}

So, factor 10^{n }out from all the numbers.

**Step 3 : **

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

**Example 1 :**

Evaluate :

(1.328 x 10^{7}) + (2.034 x 10^{5})

Give your answer in scientific notation.

**Solution :**

(1.328 x 10^{7}) + (2.034 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.

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

Then,

= (1.328 x 10^{7}) + (0.02034 x 10^{7})

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

So, factor 10^{7} out from the given numbers.

= (1.328 + 0.02034) x 10^{7}

= 1.34834 x 10^{7}

The above number is in scientific notation.

Therefore,

(1.328 x 10^{7}) + (2.034 x 10^{5}) = 1.34834 x 10^{7}

**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 10^{2} 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 10^{8}) + (338.2 x 10^{5}) - (6.1 x 10^{7})

**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 10^{8} = 0.723 x 10^{8}

338.2 x 10^{5} = 0.3382 x 10^{8}

6.1 x 10^{7 }= 0.61 x 10^{8}

**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 10^{8} = 4.512 x 10^{7}

**Method 2 :**

**Step 1 : **

First, write each number in standard notation.

0.723 x 10^{8} = 72,300,000

338.2 x 10^{5} = 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 10^{7}

After having gone through the stuff given above, we hope that the students would have understood how to add ans subtract numbers in scientific notation".

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