Question

How many centimetres are there in $1.23\times {{10}^{-6}}$kilometres?

Answer 1

Hint: We need to first find out the relation between the two units’ centimetres and kilometres to solve this problem. The problem can be solved easily in two steps. In step1, we find out the relation between the unit’s km and cm. In step2, we convert the given value in km to cm.

Complete step by step solution:
A centimetre is a unit of measurement for measuring the length of an object.
It is denoted by cm.
It is equal to one hundredth of a meter.
A meter is a basic unit of length and is also considered as a fundamental unit length.
It is denoted by m.
It is equal to a hundred centimetres.
A kilometre is a unit of measurement which is equal to thousand meters.
It is denoted by km.
The step 1 is given by,
We need to find the relation between km and cm.
A centimetre is equal to one hundredth of a meter.
$\Rightarrow 1cm=\dfrac{1}{100}m$
$\Rightarrow 1m=100cm$
Writing the above expression in powers of 10,
we get,
$\Rightarrow 1m={{10}^{2}}cm$
A kilometre is numerically equal to thousand meters.
$\Rightarrow 1km=1000m$
Writing the above expression in powers of 10,
we get,
$\Rightarrow 1km={{10}^{3}}m$
The above expression can be written as,
$\Rightarrow 1km={{10}^{3}}\times 1m$
From the above,
we know that $1m={{10}^{2}}cm$
Substituting the same,
we get,
$\Rightarrow 1km={{10}^{2}}\times {{10}^{3}}cm$
From the law of exponents,
We know that,
$\Rightarrow {{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
Applying the law of exponents,
we get,
$\Rightarrow 1km={{10}^{2+3}}cm$
$\Rightarrow 1km={{10}^{5}}cm$
The step 2 is given by,
We need to express $1.23\times {{10}^{-6}}$km in cm.
The above numerical can be also expressed as,
$\Rightarrow 1.23\times {{10}^{-6}}\times 1km$
From step1,
we know that $1km={{10}^{5}}cm$
Substituting the same,
$\Rightarrow 1.23\times {{10}^{-6}}\times {{10}^{5}}cm$
From the law of exponents,
We know that,
$\Rightarrow {{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
Applying the law of exponents,
we get,
$\Rightarrow 1.23\times {{10}^{-6+5}}cm$
$\Rightarrow 1.23\times {{10}^{-1}}cm$
Representing the above numerical in decimal form,
$\Rightarrow 0.123cm$
Therefore, the value
$1.23\times {{10}^{-6}}km$
can be expressed as 0.123cm.

Note: We need to know the basic laws of exponents to solve the problem easily. We should remember the conversions of one metric unit to the other to solve the given question in less time and to get a precise result.