Question

A calculator can perform a single calculation in about 0.0000034 sec. How much time will it take to calculate 4000 such calculations? Express your answer in scientific notation.

Answer 1

Hint: Here in the given question the number is very small, we can denote the number in the form of exponent as \[3.4 \times {10^{ - 6}}\]. To calculate the time taken by the calculator to perform 4000 calculations in \[3.4 \times {10^{ - 6}}\] can be calculated by using the unitary method. Unitary method is used to find the value of an individual unit and then multiply that value to the required value to be found.

Complete step-by-step answer:
The main thing need to be understood is to shift the decimal point wisely. When we move the decimal place towards the right the exponent is a negative number and when the decimal place moves towards the left it gets a positive number.
From the question given the time taken to calculate is 0.0000034 sec for a single calculation, therefore the given number can be expressed in the standard form as \[3.4 \times {10^{ - 6}}\].
To perform 1 calculation = 1 \[ \times \] \[3.4 \times {10^{ - 6}}\]
To find the time taken to perform 4000 such calculations we need to multiply the number of calculations with the time taken for 1 calculation as follows
\[ \Rightarrow \] \[4000 \times 3.4 \times {10^{ - 6}}\]
In the next step of simplification we will first multiply just the numerical part and retain the exponential part as it is.
\[ \Rightarrow \]\[13600 \times {10^{ - 6}}\] is the time required to calculate 4000 calculations
The end answer should be represented in scientific notation as follows
\[ \Rightarrow \] \[1.36 \times {10^4} \times {10^{ - 6}}\]
Using laws of exponents \[{a^m} \times {a^n} = {a^{m + n}}\]
\[ \Rightarrow \] \[1.36 \times {10^{4 + (}}^{ - 6)}\]
Simplifying the powers
\[ \Rightarrow \]\[1.36 \times {10^4}^{ - 6}\]
\[ \Rightarrow \] \[1.36 \times {10^{ - 2}}\] is the time taken to calculate 4000 calculations.
So, the correct answer is “ \[1.36 \times {10^{ - 2}}\]”.

Note: The given question deals with the combination of exponents and unitary method. It is must to know about the laws of exponents to solve them a few laws are \[{a^m} \times {b^m} = {(ab)^m}\], \[\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}\], \[\dfrac{{{a^m}}}{{{a^n}}} = {\left( a \right)^{m - n}}\], \[\dfrac{1}{{{a^m}}} = {a^{ - m}}\]. Unitary method is mostly used to solve the time and work, speed and distance related problems.