Question

How do you work out the value of \[\left( {7.5 \times {{10}^4}} \right) \times \left( {2.5 \times {{10}^3}} \right)\] in standard form?

Answer 1

Hint: We are given the numbers in which there is a power of 10 is mentioned. Writing a number in standard form is nothing but combining the same pattern of number and writing them in one form. Or we can say simplify the given expression as much as possible. Here we are having both the brackets with 10 to some power. So we can simplify them using the rules of indices. Also the decimal numbers can be multiplied and written in standard form. Then the power of 10 will be used to multiply with the given number so that the power of 10 is used totally.

Complete step-by-step answer:
Given that,
 \[\left( {7.5 \times {{10}^4}} \right) \times \left( {2.5 \times {{10}^3}} \right)\]
Now let’s write the numbers of same form separately,
 \[ = 7.5 \times 2.5 \times {10^4} \times {10^3}\]
First multiply the decimal numbers and place the decimal also,
 \[ = 18.75 \times {10^4} \times {10^3}\]
Now we will work on the powers of 10.
We know that if base is same then the powers can be added such that \[{a^m} \times {a^n} = {a^{m + n}}\]
So we can write,
 \[ = 18.75 \times {10^{4 + 3}}\]
Now adding the powers,
 \[ = 18.75 \times {10^7}\]
Now this is the answer so obtained. But we are asked to write it in standard form. So we will use the power of 10 to write the answer in standard form. We can split the power as
 \[ = 18.75 \times {10^2} \times {10^5}\]
Now we will use the powers one by one.
Using a second power of 10 that is 100 we removed the decimal from the number.
 \[ = 1875 \times {10^5}\]
Now we will use the remaining power.
 \[ = 1875 \times 100000\]
Now just multiply the numbers.
 \[ = 187500000\]
This is our final and answer in standard form.
So, the correct answer is “187500000”.

Note: Note that this form given to us is generally known as the scientific form of writing a number. This is used to write long distances or weights in short form. Power of 10 is the main part here. Note that the number of zeros added is nothing but the number in the power of 10. That is writing those numbers of zeros with 1.
Scientific form : \[2.5 \times {10^3}\]
Standard form : \[2500\]