Question

How do you simplify $\left( {0.4 \times {{10}^{ - 6}}} \right)\left( {0.7 \times {{10}^{ - 2}}} \right)$ and write the answer in scientific notation?

Answer 1

Hint: In scientific notation, a number is written as a product of a number lying between $1$ and $10$, and a power of $10$.
An example of scientific notation is $N \times {10^m}$ where $N$ lies between $1$ and $10$, and involves only significant figures. Scientific notation is used to express too large or too small numbers in the decimal form for a convenient expression. We are given numbers that are written in scientific notation and we have to multiply them.

Complete step by step solution:
We have to simplify $\left( {0.4 \times {{10}^{ - 6}}} \right)\left( {0.7 \times {{10}^{ - 2}}} \right)$ and find the answer in scientific notation.
So, multiply the numbers and add the power of $10$.
So, multiplying $0.4 \times {10^{ - 6}}$ with $0.7 \times {10^{ - 2}}$, we get
$0.4 \times {10^{ - 6}} \times 0.7 \times {10^{ - 2}} = 0.28 \times {10^{ - 8}}$
We know that the number of digits that is moved from the decimal point (either left or right) is given by the power of $10$ that is equal to ${10^n}$. If the decimal point is moved to the left, then the exponent $n$ is positive and if the decimal point is moved to the right, then the exponent $n$ is negative.
Now to right the result in scientific notation, we need to shift decimal in $0.28$, one point to right (to make only one digit to left of decimal point), i.e., multiply by $10$ and hence we should also multiply it by ${10^{ - 1}}$.
$\left( {0.4 \times {{10}^{ - 6}}} \right)\left( {0.7 \times {{10}^{ - 2}}} \right) = 2.8 \times {10^{ - 1}} \times {10^{ - 8}} = 2.8 \times {10^{ - 9}}$
Final solution: Therefore, $\left( {0.4 \times {{10}^{ - 6}}} \right)\left( {0.7 \times {{10}^{ - 2}}} \right) = 2.8 \times {10^{ - 9}}$.

Note:
It can also be solved by using this approach:
To write a number in scientific notation we multiply and divide the given number with the power of $10$ such that the power is equal to (total number of digits$ - 1$), this way the decimal point is placed after the first digit. The scientific notation should contain only significant figures so the digits after the decimal point are rounded off.
Now, the scientific notation can be easily converted into standard form by multiplying the decimal number with the power of $10$. While writing the given number in scientific notation, we see that there is only one significant figure so no decimal will be involved, thus the calculation is easy. This way we can solve similar questions.