OPERATIONS WITH SCIENTIFIC NOTATION

Example 1 :

Simplify the expression given below. 

(4 x 10⁵) + (0.1 x 10⁷)

Solution : 

Method 1 :

Step 1 : 

In the given numbers, the highest power of 10 is 7. 

So, write each number with 10 power 7.

4 x 10⁵  =  0.04 x 10⁷

0.1 x 10⁷  =  0.1 x 10⁷

Step 2 : 

Add the multipliers for each number.

0.04 + 0.1  =  0.14

Step 3 : 

Write the final answer in scientific notation :

0.14 x 10⁷  =  1.4 x 10⁶

Method 2 :

Step 1 : 

First, write each number in standard notation.

4 x 10⁵  =  400,000

0.1 x 10⁷  =  1,000,000

Step 2 : 

Find the sum of the numbers in standard notation.

400,000 + 1,000,000  =  1,400,000

Step 3 : 

Write the final answer in scientific notation :

1,400,000  =  1.4 x 10⁶

Example 2 : 

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

Example 3 :

Multiply :

(3.2 x 10⁵) x (26.7 x 10³)

Solution :

Step 1 :

Find the product of multipliers for each number. 

3.2 x 26.7  =  85.44

Step 2 :

Find the product of powers of 10.

10⁵ x 10³  =  10⁸

Step 3 :

Combine the results of step 1 and step 2 to write the final answer in scientific notation.

85.44 x 10⁸  =  8.544 x 10⁹

Example 4 :

When the Sun makes an orbit around the center of the Milky Way, it travels 2.025 × 10¹⁴ kilometers. The orbit takes 225 million years. At what rate does the Sun travel? Write your answer in scientific notation.

Solution :

Key points : 

The answer is the number of kilometers per year that the Sun travels around the Milky Way.

Set up a division problem using

Rate  =  Distance / Time

to represent the situation.

Step 1 : 

Substitute the values from the problem into the Rate formula. 

Step 2 : 

Write the expression for rate with years in scientific notation.

That is, 225 million  =  2.25 x 10⁸.

Then, we have

Step 3 :

Find the quotient by dividing the decimals and using the laws of exponents.

Divide the multipliers.

2.025 ÷  2.25  =  0.9

Divide the powers of 10.

10¹⁴ ÷ 10⁸  =  10^14-8

10¹⁴ ÷ 10⁸  =  10⁶

Step 4 :

Combine the answers to write the rate in scientific notation.

0.9 x 10⁶  =  9.0 x 10⁵

Justify and Evaluate : 

Use estimation to check the reasonableness of your answer.

9.0 x 10⁵ is close 10⁶, so the answer is reasonable.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us : 

v4formath@gmail.com

We always appreciate your feedback. 

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6