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Mathematics Question on Use of Exponents to Express Small Numbers in Standard Form

Express the following numbers in standard form.
(i) 0.0000000000085
(ii) 0.00000000000942
(iii) 6020000000000000
(iv) 0.00000000837
(v) 31860000000

Answer

(i) 0.0000000000085
To convert this small number into the standard form we need to move the decimal to its right by 12 steps.
=8510000000000000= \frac{85}{10000000000000}

=8.5×101013= 8.5 × \frac{10}{10¹³}
=8.5×10−13×10= 8.5 × 10^{-13} × 10
=8.5×10−12= 8.5 × 10^{-12}
∴ The standard form of a given number is 8.5×10−128.5 × 10^{-12}

(ii) 0.00000000000942
=942100000000000000= \frac{942}{100000000000000}

=9.42×1021014= 9.42 × \frac{10^2 }{ 10^{14}}
=9.42×10−14×102= 9.42 × 10^{−14} × 10^2
=9.42×10−12= 9.42 × 10^{−12}
∴ The standard form of a given number is 9.42×10−129.42 × 10^{−12}

(iii) 6020000000000000
= 6.02 × 102 × 10000000000000
= 6.02 × 102 × 1013
= 6.02 × 1015
∴ The standard form of the given number is 6.2 × 1015

(iv) 0.00000000837
=837100000000000= \frac{837 }{ 100000000000}

=8.37×1021011= 8.37 × \frac{10^2 }{ 10^{11}}

= 8.37 × 10−11×102
= 8.37 × 10−9
∴ The standard form of a given number is 8.37 × 10−9

(v) 31860000000
= 3.186 × 103 × 107
= 3.186 ×1010
∴ The standard form of the given number is 3.186 × 1010