Square root and Approximations
Scientific and Standard Notations
Linear inequalities

Operations on Scientific Notations worksheet

Addition and subtraction in scientific notation:

While performing basic arithmetic operations like addition, subtraction, multiplication or division on   numbers written in scientific notation, we need to be careful and should follow some rules to reach the correct answer.

While adding and subtracting the numbers in scientific notation, exponent part of all the numbers must be same. If it is not same , then first make it same and then combine the numbers.

Don’t forget to write the final answer in scientific notation.

When exponent part  is same-

Example1: Combine \dpi{120} {\color{Red} 4.52\times 10^{3}}   and  \dpi{120} {\color{Red} 8.3\times 10^{3}}

Solution: Here we see that exponent part of both numbers is same (\dpi{120} 10^{3}). So we just need to combine the coefficients of these two numbers.

\dpi{120} 4.52\times 10^{3} + \dpi{120} 8.3\times 10^{3}   =  \dpi{120} (4.52+8.3)\times 10^{3}

= \dpi{120} 12.82\times 10^{3}

As the coefficient part can’t exceed 10, so we need to convert this  result  into scientific notation.

=  \dpi{120} 1.282\times 10^{4}

 

When exponent part is not same-

Example2: Simplify the following expression:   \dpi{120} {\color{Red} 8\times 10^{8}- 5\times 10 ^{6}}

Solution: As the exponent part is not same, we make them same first.

Again you can do it two ways. Either lower the exponent  \dpi{120} 10^{8}  to \dpi{120} 10^{6}   or  you can raise the exponent \dpi{120} 10^{6}   to  \dpi{120} 10^{8}

First way:  raise the exponent \dpi{120} 10^{6}   to  \dpi{120} 10^{8}    

As we know, when decimal is moved to left, exponent is positive (increased). We need to raise the exponent by 2 so decimal is moved towards left by 2 places and we get \dpi{120} 5\times 10^{6} = 0.05\times 10^{2}\times 10 ^{6}

= \dpi{120} 0.05\times 10^{8}

\dpi{120} 8\times 10^{8}- 5\times 10 ^{6}   =  \dpi{120} 8\times 10^{8}- 0.05\times 10 ^{8}

= \dpi{120} (8-0.05)\times 10^{8}

=  \dpi{120} 7.95\times 10^{8}

Second way:  Either lower the exponent  \dpi{120} 10^{8}  to \dpi{120} 10^{6}

When decimal is moved to right, exponent is negative (decreased). We need to lower the exponent by 2 so decimal is moved towards right by 2 places  and we get    \dpi{120} 8\times 10 ^{8} = 8\times 10^{2}\times 10^{6} = 800\times 10^{6}

\dpi{120} 8\times 10^{8}- 5\times 10 ^{6}  = \dpi{120} 800\times 10^{6}- 5\times 10 ^{6}

=  \dpi{120} 795\times 10^{6}

But this result is not in scientific notation. To get it in scientific notation decimal should be placed after first significant digit.

So final answer is   \dpi{120} 7.95\times 10^{8}

 

Example3. Simplify the expression  \dpi{120} {\color{Red} 9\times 10^{5} + 6\times 10 ^{7}}

Solution:   \dpi{120} 9\times 10^{5} + 6\times 10 ^{7}  =   \dpi{120} 9\times 10^{5} + 6\times 10 ^{2}\times 10^{5}

=   \dpi{120} 9\times 10^{5} + 600\times 10^{5}

= \dpi{120} (9+600)\times 10^{5}

=  \dpi{120} 609 \times 10^{5}

Then we convert it into scientific notation and get answer as  \dpi{120} 6.09 \times 10^{7}

 

Example4. Simplify the expression  \dpi{120} {\color{Red} 9\times 10^{5} - 6\times 10 ^{7}}

Solution:   \dpi{120} 9\times 10^{5} - 6\times 10 ^{7}   =   \dpi{120} 9\times 10^{5} - 6\times 10 ^{2}\times 10^{5}

=   \dpi{120} 9\times 10^{5} - 600\times 10^{5}

=  \dpi{120} (9-600)\times 10^{5}

=    \dpi{120} -591 \times 10^{5}

Rewriting it as scientific notation we get the final answer as   \dpi{120} -5.91\times 10^{7}

 

 

Practice problems:

Simplify each problem and write the answer in scientific notation.

  1. \dpi{120} 1\times 10^{6} - 8\times 10 ^{4}
  2. \dpi{120} 7\times 10^{7} + 3\times 10 ^{4}
  3. \dpi{120} 4\times 10^{8} - 9\times 10 ^{7}
  4.  \dpi{120} 3\times 10^{-2} - 6\times 10 ^{-5}

 

Answers:

1) \dpi{120} 9.2\times 10^{5}

2) \dpi{120} 7.003\times 10 ^{7}

3) \dpi{120} 3.1\times 10^{8}

4) \dpi{120} 2.994\times 10^{-2}

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