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: As, it is a question from exponent and power, here in these question the power of the base, i.e., 10 is to be taken into consideration, as whenever any base has power of negative number, then the reciprocal of the base is taken, as by taking the reciprocal of any number then its power sign is changed. We need to expand the base with the help of the power.

Complete step-by-step answer:
Given,
I.$ 3.02 \times {10^{ - 6}} $ .
We need to find the usual form of the above expression.
To evaluate this value we should follow the law of exponent.
As, $ {a^{ - n}} = \dfrac{1}{{{a^n}}} $ and $ \dfrac{1}{{{b^{ - n}}}} = {b^n} $ .
Expression $ 3.02 \times {10^{ - 6}} $ consist of one term as $ {10^{ - 6}} $ , so it can be written as,
$\Rightarrow {10^{ - 6}} = \dfrac{1}{{{{10}^6}}} $ , here $ {10^6} $ means, there will be six zeros after 1, i.e., $ \dfrac{1}{{{{10}^6}}} = \dfrac{1}{{1000000}} $ .
Now, we have to simplify the expression $ 3.02 \times {10^{ - 6}} $ as follows,
 $ 3.02 \times {10^{ - 6}} = 3.02 \times \dfrac{1}{{1000000}} $
We need to observe one thing here as there are six zero’s after one, therefore by dividing $ 3.02 $ by 1000000.
 $ 3.02 \times {10^{ - 6}} = 3.02 \times \dfrac{1}{{1000000}} = 0.000000302 $
Therefore, the usual form of $ 3.02 \times {10^{ - 6}} $ will be 0.000000302.

II.Given, $ 4.5 \times {10^4} $ .
We need to find the usual form of the above expression.
As, $ {10^4} $ can be expanded as 10000.
So, we need to multiply $ 4.5 $ with 10000.
 $\Rightarrow 4.5 \times {10^4} = 4.5 \times 10000 = 45000 $
We need to keep in mind that one zero will be gone for removing the decimal, so we will be left with 1000.
Therefore, the usual form of $ 4.5 \times {10^4} $ will be 45000.
Given, $ 3 \times {10^{ - 8}} $
We need to find the usual form of the above expression.
To evaluate this value we should follow the law of exponent.
As, $ {a^{ - n}} = \dfrac{1}{{{a^n}}} $ and $ \dfrac{1}{{{b^{ - n}}}} = {b^n} $ .
Expression $ 3 \times {10^{ - 8}} $ consist of one term as $ {10^{ - 8}} $ , so it can be written as,
 $ {10^{ - 8}} = \dfrac{1}{{{{10}^8}}} $ , here $ {10^8} $ means, there will be eight zeros after 1, i.e., $ \dfrac{1}{{{{10}^8}}} = \dfrac{1}{{100000000}} $ .

III.Now, we have to simplify the expression $ 3 \times {10^{ - 8}} $ as follows,
 $ 3 \times {10^{ - 8}} = 3 \times \dfrac{1}{{100000000}} $
We need to observe one thing here as there are eight zeros after one, therefore by dividing 3 by 100000000, there will be eight zero ahead of digit three.
 $ 3 \times {10^{ - 8}} = 3 \times \dfrac{1}{{100000000}} = 0.000000003 $
Therefore, the usual form of $ 3 \times {10^{ - 8}} $ will be 0.000000003.

VI.Given, $ 1.0001 \times {10^9} $
We need to find the usual form of the above expression.
As, $ {10^9} $ can be expanded as 1000000000.
So, we need to multiply $ 1.0001 $ with 1000000000.
 $\Rightarrow 1.0001 \times {10^9} = 1.0001 \times 1000000000 = 1000100000 $
We need to keep in mind that there are four digits after decimal therefore four zeros will be gone for removing the decimal.
Therefore, the usual form of $ 1.0001 \times {10^9} $ will be 1000100000.

V.Given, $ 5.8 \times {10^{12}} $
We need to find the usual form of the above expression.
As, $ {10^9} $ can be expanded as 1000000000.
So, we need to multiply $ 1.0001 $ with 1000000000.
 $\Rightarrow 1.0001 \times {10^9} = 1.0001 \times 1000000000 = 1000100000 $
We need to keep in mind that there are four digits after decimal therefore four zeros will be gone for removing the decimal.
Therefore, the usual form of $ 1.0001 \times {10^9} $ will be 1000100000.

VI..Given $ 3.61492 \times {10^6} $
We need to find the usual form of the above expression.
As, $ {10^6} $ can be expanded as 1000000.
So, we need to multiply $ 3.61492 $ with 1000000.
\[\Rightarrow 3.61492 \times {10^6} = 3.61492 \times 1000000 = 3614920\]
We need to keep in mind that there are five digits after decimal therefore five zeros will be gone for removing the decimal.
Therefore, the usual form of \[3.61492 \times {10^6}\] will be 3614920.

Note: This question is of the concept power and exponent. Power or exponent of any number is the number of times any particular number is being multiplied by it. Whenever in any expression there is involvement of two things base and power that number is called an exponential number.