Question

Write the following using base \[10\] and exponents:
\[\left( {\text{i}} \right)12345\]
\[\left( {{\text{ii}}} \right)1010.0101\]
\[\left( {{\text{iii}}} \right)0.1020304\]

Answer 1

Hint:Here, we see they are of numbers, want to write it as base \[10\] and exponents. Exponent is a mathematical operation, which of the form \[{a^x}\].Base \[10\] is a tool determined by the number of zeros present in the format.
For example, I choose \[40\] we can write it as \[4 \times 10 = {2^2} \times 10\], where \[{2^2}\] is exponents and \[10\] is number of zeros it have, it shows that we want to write the numbers as in the format multiples of ones, tens, hundreds, thousands, etc.
Here ones = single digit
Tens = \[{\text{digit}} \times 10\]
Hundreds = \[{\text{digit}} \times 100 = {\text{digit}} \times {10^2}\]
Thousands= \[{\text{digit}} \times 1000 = {\text{digit}} \times {10^3}\] etc.
In the case of decimal points as we write in two ways, they are the number given in the left side, we split it into multiplication and a number given in the right side , we can split by using division.

Complete step-by-step answer:
i) $12345$
We write \[12345\] as words, twelve thousand three hundred and forty five.
\[12345 = 10000 + 2000 + 300 + 40 + 5\]
Also, we split this number \[12345\] as
\[12345 = 1 \times 10000 + 2 \times 1000 + 3 \times 100 + 4 \times 10 + 5\]
We write in exponent form as
\[12345 = {10^4} + 2 \times {10^3} + 3 \times {10^2} + 4 \times 10 + 5\].

ii) $1010.0101$
In this question is in decimal form, so we split it into
\[1010.0101 = 1010 + 0.0101\]
Also we split,
\[1010.0101 = 1 \times 1000 + 0 \times 100 + 1 \times 10 + 0 + \dfrac{0}{{10}} + \dfrac{1}{{{{10}^2}}} + \dfrac{0}{{{{10}^3}}} + \dfrac{1}{{{{10}^4}}}\]
Simplify it,
\[1010.0101 = 1 \times 1000 + 0 + 1 \times 10 + 0 + 0 + \dfrac{1}{{{{10}^2}}} + 0 + \dfrac{1}{{{{10}^4}}}\]
On some simplification we can write.
\[1010.0101 = 1 \times 1000 + 1 \times 10 + \dfrac{1}{{{{10}^2}}} + \dfrac{1}{{{{10}^4}}}\]
Multiply the left side terms, we get
\[1010.0101 = 1000 + 10 + \dfrac{1}{{{{10}^2}}} + \dfrac{1}{{{{10}^4}}}\]
We put into the powers of \[1000\],
\[1010.0101 = {10^3} + 10 + \dfrac{1}{{{{10}^2}}} + \dfrac{1}{{{{10}^4}}}\]

iii) $0.1020304$
We write \[0.1020304\] as
$0.1020304 = \dfrac{1}{{10}} + \dfrac{0}{{{{10}^2}}} + \dfrac{2}{{{{10}^3}}} + \dfrac{0}{{{{10}^4}}} + \dfrac{3}{{{{10}^5}}} + \dfrac{0}{{{{10}^6}}} + \dfrac{4}{{{{10}^7}}}$
Cancelled the term of 0 in the numerator, we get
$0.1020304 = \dfrac{1}{{10}} + \dfrac{2}{{{{10}^3}}} + \dfrac{3}{{{{10}^5}}} + \dfrac{4}{{{{10}^7}}}$

Additional Information:
Law of exponents: Multiplying the power with same base
${a^m} \times {a^n} = {a^{m + n}}$
${\left( {\dfrac{a}{b}} \right)^m} \times {\left( {\dfrac{a}{b}} \right)^n} = {\left( {\dfrac{a}{b}} \right)^{m + n}}$

Note:Every number has a base \[10\] and exponent, we can write a single digit (ones digit) as that format.
E.g.: we can write \[1\] has \[1 \times {10^0}\] where \[{10^0} = 1\], so that the single digit (ones digit) is simply write as single digit (one`s digit).
Also we can write \[\dfrac{1}{{10}} = {10^{ - 1}}\] , \[\dfrac{2}{{10}} = {10^{ - 2}}\] , etc.,
Further, \[\dfrac{0}{{{\text{anything}}}} = 0\] which is \[\dfrac{0}{{10}} = 0\] , \[\dfrac{0}{{{{10}^3}}} = 0\] and so.,
Above mentioned properties are important to solve problems, it creates a logic analyser of the number which format it presents. The exponents and base \[10\] properties are mostly used as mathematical tools. It mostly represents the power, some physical and chemical value mentioned by above way.