Question

How do you write the answer in scientific notation given \[\dfrac{{{{\left( {1.2 \times {{10}^{ - 4}}} \right)}^2}}}{{\left( {9.0 \times {{10}^5}} \right)\left( {1.6 \times {{10}^{ - 8}}} \right)}}\]?

Answer 1

Hint: Here in this question, we have to write the scientific notation of a given fraction in simplest form. This can be solved by using the law of indices first we can solve the exponent term one by one and later multiply or divide the numerical term on further simplification using basic arithmetic operation we get the required solution.

Complete step by step solution:
This question comes under the topic number system. The number can be expressed in many different ways but it has the same value. The number can be written in the words form or in the numeral form or in the scientific form etc.
Consider the given fraction:
\[ \Rightarrow \,\,\,\dfrac{{{{\left( {1.2 \times {{10}^{ - 4}}} \right)}^2}}}{{\left( {9.0 \times {{10}^5}} \right)\left( {1.6 \times {{10}^{ - 8}}} \right)}}\]----------(1)
Now recall the, some exponent rules
Assume that a and b are nonzero real numbers, and m and n are many integers.
Zero property of exponent
\[{b^0} = 1\]
Negative property of exponent
\[{b^{ - n}} = \dfrac{1}{{{b^n}}}\] or \[\dfrac{1}{{{b^{ - n}}}} = {b^n}\]
Product property of exponent
\[\left( {{b^m}} \right)\left( {{b^n}} \right) = {b^{m + n}}\]
Quotient property of exponent
\[\dfrac{{{b^m}}}{{{b^n}}} = {b^{m - n}}\]
Power of a power property of exponent
\[{\left( {{b^m}} \right)^n} = {b^{m\,n}}\]
Power of a product property of exponent
\[{\left( {ab} \right)^m} = {a^m}{b^m}\]
Power of a quotient property of exponent
\[{\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}\]
Rewrite the numerator of equation (1) using the Power of a product property of exponent, then
\[ \Rightarrow \,\,\,\dfrac{{{{\left( {1.2} \right)}^2} \times {{\left( {{{10}^{ - 4}}} \right)}^2}}}{{9.0 \times {{10}^5} \times 1.6 \times {{10}^{ - 8}}}}\]
Now apply the fifth property of exponent in numerator and third property of exponent in denominator, then
 \[ \Rightarrow \,\,\,\dfrac{{1.44 \times {{10}^{ - 4 \times 2}}}}{{14.4 \times {{10}^{5 - 8}}}}\]
On simplification, we have
\[ \Rightarrow \,\,\,\dfrac{{1 \times {{10}^{ - 8}}}}{{10 \times {{10}^{ - 3}}}}\]
Again, by the third property denominator can be written as
\[ \Rightarrow \,\,\,\dfrac{{{{10}^{ - 8}}}}{{{{10}^{1 - 3}}}}\]
\[ \Rightarrow \,\,\,\dfrac{{{{10}^{ - 8}}}}{{{{10}^{ - 2}}}}\]
Now, using the second property of exponent then
\[ \Rightarrow \,\,\,{10^{ - 8}} \times {10^2}\]
Again, by the third property of exponent
\[ \Rightarrow \,\,\,{10^{ - 8 + 2}}\]
On simplification, we get
\[ \Rightarrow \,\,\,{10^{ - 6}}\]

Hence, the simplest Scientific notation of given fraction is \[{10^{ - 6}}\]

Note: There are different many to express or to write the number or to express the number. The exponent form is defined as the number of times the number is multiplied by the number itself. The place value table of numbers should be known to place the number in particular that position.