Question

Express the following numbers in the standard form:
$
  {\text{(a) }}0.0000000000085 \\
  {\text{(b) }}0.00000000000942 \\
  {\text{(c) }}6020000000000000 \\
  {\text{(d) }}0.00000000837 \\
  {\text{(e) }}31860000000 \\
 $

Answer 1

Hint: Here we have to use the concept of leading zeroes, non-leading and trailing zeroes to assign the powers of 10 , that is either negative power or positive power.

Complete step-by-step answer:

(a). \[0.0000000000085\]
Here, we have $13$numbers after the decimal including $11$ zeros and $2$ non-zero numbers. And we know that we use zeros as place holders at the end of a number to denote insignificant digits. So, zero has no value after decimal. That is, the Leading zeros (zeros before non-zero numbers) are not significant and can be written as the negative power of $10$. Therefore, we can write the number \[0.0000000000085\]as
\[ = \;85 \times {10^{ - 13}}\]
\[ = \;8.5 \times {10^{ - 12}}\]

(b). $0.00000000000942$
Here, we have $14$numbers after the decimal including $11$ zeros and $3$ non-zero numbers. And we know that we use zeros as place holders at the end of a number to denote insignificant digits. So, zero has no value after decimal. That is, the Leading zeros (zeros before non-zero numbers) are not significant and can be written as the negative power of $10$. Therefore, we can write the number $0.00000000000942$as
\[\; = 942 \times {10^{ - 14}}\]
\[\; = 9.42 \times {10^{ - 12}}\]

(c). $6020000000000000$
Here, we have $3$ non-zero numbers and $13$ trailing zeros. Trailing zeros are a sequence of zeros in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. Also, when the zeros are written before or without the decimal point, the number of zeros is written as the positive power of $10$. Thus, here the given number can be represented as:
\[ = \;602 \times {10^{13}}\]\[\]
\[ = \;6.02 \times {10^{15}}\]

(d). ${\text{ }}0.00000000837$
Here, we have $11$numbers after the decimal including $8$ zeros and $3$ non-zero numbers. And we know that we use zeros as place holders at the end of a number to denote insignificant digits. So, zero has no value after decimal. That is, the Leading zeros (zeros before non-zero numbers) are not significant. Therefore, we can write the number ${\text{ }}0.00000000837$as
\[ = \;837 \times {10^{ - 11}}\]
\[ = \;8.37 \times {10^{ - 9}}\]

(e). ${\text{ }}31860000000$
Here, we have $4$non-zero numbers and $7$ trailing zeros. Trailing zeros are a sequence of zeros in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. And these are represented as the positive power of $10$.
\[ = \;3186 \times {10^7}\]
\[ = \;3.186 \times {10^{10}}\]

Note: In these types of questions, the knowledge of significant numbers is must because it will help to find the standard form.