Question

Express the following numbers in standard form:
(a) \[0.0000000000085\]
(b) \[0.00000000000942\]
(c) 6020000000000000
(d) \[0.00000000837\]
(e) 31860000000

Answer 1

Hint:
Here, we need to express the given numbers in standard form. A number is said to be in standard form if it is written as \[a \times {10^b}\], where \[a\] is a number between 1 and 10. We will rewrite the given numbers in the form \[a \times {10^b}\] using rules of exponents.

Formula Used:
We will use the following formulas:
\[{a^m} \times {a^n} = {a^{m + n}}\].
The number \[\dfrac{1}{{{a^b}}}\] can be written as the number \[{a^{ - b}}\] with a negative exponent.

Complete step by step solution:
We need to express the given numbers in standard form.
A number is said to be in standard form if it is written as \[a \times {10^b}\], where \[a\] is a number between 1 and 10 (1 included, 10 excluded).
We will rewrite the given numbers in the form \[a \times {10^b}\] using rules of exponents.
(a)
The given number is \[0.0000000000085\].
Representing the decimal number as a fraction, we get
\[ \Rightarrow 0.0000000000085 = \dfrac{{85}}{{10000000000000}}\]
Rewriting the numerator as a number between 1 and 10, we get
\[ \Rightarrow 0.0000000000085 = \dfrac{{8.5}}{{1000000000000}}\]
The number of zeroes in the denominator is 12.
The number 1000000000000 can be written as the product \[10 \times 10 \times \ldots \ldots \] 12 times.
Thus, the number 1000000000000 can be written as the number \[{10^{1 + 1 + 1 + \ldots \ldots {\rm{ 12 times}}}}\].
Simplifying the expression, we get
\[1000000000000 = {10^{12}}\]
Thus, we get
\[ \Rightarrow 0.0000000000085 = \dfrac{{8.5}}{{{{10}^{12}}}}\]
Rewriting the expression, we get
\[ \Rightarrow 0.0000000000085 = 8.5 \times \dfrac{1}{{{{10}^{12}}}}\]
The number \[\dfrac{1}{{{a^b}}}\] can be written as the number \[{a^{ - b}}\] with a negative exponent.
Therefore, we get
\[ \Rightarrow 0.0000000000085 = 8.5 \times {10^{ - 12}}\]
Thus, we have written the number \[0.0000000000085\] in the standard form \[8.5 \times {10^{ - 12}}\].
 (b)
The given number is \[0.00000000000942\].
Representing the decimal number as a fraction, we get
\[ \Rightarrow 0.00000000000942 = \dfrac{{942}}{{100000000000000}}\]
Rewriting the numerator as a number between 1 and 10, we get
\[ \Rightarrow 0.00000000000942 = \dfrac{{9.42}}{{1000000000000}}\]
The number of zeroes in the denominator is 12.
The number 1000000000000 can be written as the product \[10 \times 10 \times \ldots \ldots \] 12 times.
Thus, the number 1000000000000 can be written as the number \[{10^{1 + 1 + 1 + \ldots \ldots {\rm{ 12 times}}}}\].
Simplifying the expression, we get
\[1000000000000 = {10^{12}}\]
Thus, we get
\[ \Rightarrow 0.00000000000942 = \dfrac{{9.42}}{{{{10}^{12}}}}\]
Rewriting the expression, we get
\[ \Rightarrow 0.00000000000942 = 9.42 \times \dfrac{1}{{{{10}^{12}}}}\]
Therefore, we get
\[ \Rightarrow 0.00000000000942 = 9.42 \times {10^{ - 12}}\]
Thus, we have written the number \[0.00000000000942\] in the standard form \[9.42 \times {10^{ - 12}}\].
(c)
The given number is 6020000000000000.
Rewriting the number, we get
\[ \Rightarrow 6020000000000000 = 602 \times 10000000000000\]
Multiplying and dividing the expression by 100, we get
\[\begin{array}{l} \Rightarrow 6020000000000000 = 602 \times 10000000000000 \times \dfrac{{100}}{{100}}\\ \Rightarrow 6020000000000000 = \dfrac{{602}}{{100}} \times 1000000000000000\end{array}\]
Rewriting the first number as a number between 1 and 10, we get
\[ \Rightarrow 6020000000000000 = 6.02 \times 1000000000000000\]
The number of zeroes in the second number is 15.
The number 1000000000000000 can be written as the product \[10 \times 10 \times \ldots \ldots \] 15 times.
Thus, the number 1000000000000000 can be written as the number \[{10^{1 + 1 + 1 + \ldots \ldots {\rm{ 15 times}}}}\].
Simplifying the expression, we get
\[1000000000000000 = {10^{15}}\]
Thus, we get
\[ \Rightarrow 6020000000000000 = 6.02 \times {10^{15}}\]
Thus, we have written the number 6020000000000000 in the standard form \[6.02 \times {10^{15}}\].
(d)
The given number is \[0.00000000837\].
Representing the decimal number as a fraction, we get
\[ \Rightarrow 0.00000000837 = \dfrac{{837}}{{100000000000}}\]
Rewriting the numerator as a number between 1 and 10, we get
\[ \Rightarrow 0.00000000837 = \dfrac{{8.37}}{{1000000000}}\]
The number of zeroes in the denominator is 9.
The number 1000000000 can be written as the product \[10 \times 10 \times \ldots \ldots \] 9 times.
Thus, the number 1000000000 can be written as the number \[{10^{1 + 1 + 1 + \ldots \ldots {\rm{ 9 times}}}}\].
Simplifying the expression, we get
\[1000000000 = {10^9}\]
Thus, we get
\[ \Rightarrow 0.00000000837 = \dfrac{{8.37}}{{{{10}^9}}}\]
Rewriting the expression, we get
\[ \Rightarrow 0.00000000837 = 8.37 \times \dfrac{1}{{{{10}^9}}}\]
Therefore, we get
\[ \Rightarrow 0.00000000837 = 8.37 \times {10^{ - 9}}\]
Thus, we have written the number \[0.00000000837\] in the standard form \[8.37 \times {10^{ - 9}}\].
(e)
The given number is 31860000000.
Rewriting the number, we get
\[ \Rightarrow 31860000000 = 3186 \times 10000000\]
Multiplying and dividing the expression by 1000, we get
\[\begin{array}{l} \Rightarrow 31860000000 = 3186 \times 10000000 \times \dfrac{{1000}}{{1000}}\\ \Rightarrow 31860000000 = \dfrac{{3186}}{{1000}} \times 10000000000\end{array}\]
Rewriting the first number as a number between 1 and 10, we get
\[ \Rightarrow 31860000000 = 3.186 \times 10000000000\]
The number of zeroes in the second number is 10.
The number 10000000000 can be written as the product \[10 \times 10 \times \ldots \ldots \] 10 times.
Thus, the number 10000000000 can be written as the number \[{10^{1 + 1 + 1 + \ldots \ldots {\rm{ 10 times}}}}\].
Simplifying the expression, we get
\[10000000000 = {10^{10}}\]
Thus, we get
\[ \Rightarrow 31860000000 = 3.186 \times {10^{10}}\]

Thus, we have written the number 31860000000 in the standard form \[3.186 \times {10^{10}}\].

Note:
The number \[{10^{ - 9}}\] is a number in exponential form. The exponential form of a number is written as \[{x^y}\], where \[x\] is called the base and \[y\] is called the exponent. \[{x^y}\] means \[x\] raised to the power of \[y\], that is \[x \times x \times x \times \ldots \ldots \ldots \] upto \[y\] times. If two numbers with exponents having the same base are multiplied, then the result is the same base, with the power as the sum of exponents. This can be written as \[{a^m} \times {a^n} = {a^{m + n}}\].