Question

Express the following numbers in usual form.
(i) \[3.02 \times {10^{ - 6}}\]
(ii) \[4.5 \times {10^4}\]
(iii) \[3 \times {10^{ - 8}}\]
(iv) \[1.0001 \times {10^9}\]
(v) \[5.8 \times {10^{12}}\]
(vi) \[3.61492 \times {10^6}\]

Answer 1

Hint:These are problems of decimal functions. To solve these we will use exponential formulae. First we will convert these questions into fraction form then we will simplify it using exponential formulae to get the answer.

Complete step-by-step answer:
(i) \[3.02 \times {10^{ - 6}}\]
We can write the decimal in fraction form as,
 $ = \dfrac{{302}}{{100}} \times {10^{ - 6}} $
 $ = \dfrac{{302}}{{{{10}^2}}} \times {10^{ - 6}} $
We can write $ {10^{ - 6}} $ as $ \dfrac{1}{{{{10}^6}}} $ ,
 $ = \dfrac{{302}}{{{{10}^2}}} \times \dfrac{1}{{{{10}^6}}} $
Multiplying the denominator using exponential formula $ {a^m} \times {a^n} = {a^{m + n}} $ we get,
 $ = \dfrac{{302}}{{{{10}^{2 + 6}}}} $
 $ = \dfrac{{302}}{{{{10}^8}}} $
Converting this into decimal form we get,
 $ = 0.00000302 $
 $ \therefore $ $ 3.02 \times {10^{ - 6}} = 0.00000302 $

(ii) \[4.5 \times {10^4}\]
We can write the decimal in fraction form as,
 $ = \dfrac{{45}}{{10}} \times {10^4} $
We can write it as,
 $ = 45 \times \dfrac{{{{10}^4}}}{{10}} $
Using exponential formula $ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $ we get,
 $ = 45 \times {10^{4 - 1}} $
 $ = 45 \times {10^3} $
Converting this into decimal form we get,
 $ = 45000 $
 $ \therefore $ $ 4.5 \times {10^4} = 45000 $

(iii) \[3 \times {10^{ - 8}}\]
We can write the decimal in fraction form as,
\[ = \dfrac{3}{{{{10}^8}}}\]
Converting this into decimal form we get,
 $ = 0.00000003 $
 $ \therefore $ $ 3 \times {10^{ - 8}} = 0.00000003 $

(iv) \[1.0001 \times {10^9}\]
We can write the decimal in fraction form as,
 $ = \dfrac{{10001}}{{10000}} \times {10^9} $
We can write 10000 as $ {10^4} $ ,
 $ = \dfrac{{10001}}{{{{10}^4}}} \times {10^9} $
 $ = 10001 \times \dfrac{{{{10}^9}}}{{{{10}^4}}} $
Multiplying the denominator using exponential formula $ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $ we get,
 $ = 10001 \times {10^{9 - 4}} $
 $ = 10001 \times {10^5} $
Converting this into decimal form we get,
 $ = 1000100000 $
 $ \therefore $ $ 1.0001 \times {10^9} = 1000100000 $

(v) \[5.8 \times {10^{12}}\]
We can write the decimal in fraction form as,
 $ = \dfrac{{58}}{{10}} \times {10^{12}} $
 $ = 58 \times \dfrac{{{{10}^{12}}}}{{10}} $
Using exponential formula $ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $ we get,
 $ = 58 \times {10^{12 - 1}} $
 $ = 58 \times {10^{11}} $
Converting this into decimal form we get,
 $ = 5800000000000 $
 $ \therefore $ $ 5.8 \times {10^{12}} = 5800000000000 $

(vi) \[3.61492 \times {10^6}\]
We can write the decimal in fraction form as,
 $ = \dfrac{{361492}}{{100000}} \times {10^6} $
We can write 100000 as $ {10^5} $ ,
 $ = \dfrac{{361492}}{{{{10}^5}}} \times {10^6} $
 $ = 361492 \times \dfrac{{{{10}^6}}}{{{{10}^5}}} $
Using exponential formula $ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $ we get,
 $ = 361492 \times {10^{6 - 5}} $
 $ = 361492 \times 10 $
Converting this into decimal form we get,
 $ = 3614920 $
 $ \therefore $ $ 3.61492 \times {10^6} = 3614920 $

Note:In algebra a decimal number can be defined as a number whose whole number part and fractional part is separated by a decimal point.A fraction represents a part of a whole.Students should remember all the formulae, properties and rules of decimal function, fraction and exponential function.