Question

How do you write \[8\times {{10}^{3}}\] in the standard notation.

Answer 1

Hint: We are given a term as \[8\times {{10}^{3}}\] and we are asked to write it in standard notation. To do so we will learn what type of notation is this and in which it is written and then we will learn about the standard notation. We will use the property that \[{{x}^{a}}=\underbrace{x\times x\times x......\times x}_{\text{a times}}\] to simplify and solve. Using this knowledge, we will be able to change \[8\times {{10}^{3}}\] into standard notation.

Complete step-by-step solution:
We are given a term as \[8\times {{10}^{3}}\] and if we look closely we can see that the term has one term as 8 and the other in the exponent of the base 10. So, we can understand that the form is a scientific notation form. So, here we are asked to change the scientific notation to the standard notation where the standard notation means to write the scientific term to the decimal form. To do so we have to understand how the scientific notation comes. If we have the decimal term in which non – zero terms is after 3 units from the decimal to the right, then the power of the exponent will rise to a negative of 3. For example, \[0.\underrightarrow{00}3\] it will become \[3\times {{10}^{-3}}.\] And if the non – zero terms is after 3 units from the decimal to the left, then the power will rise to + 3. For example, \[3\underleftarrow{000} \] it will become \[3\times {{10}^{3}}.\] Similarly, we just had to use this knowledge in the reverse order that is if the power of 10 is positive then the decimal will move to the right and if the power is negative then the decimal will move to the left. For example, \[2.0\times {{10}^{2}}\] the power is positive 2. So, the decimal will move 2 units to the right. So, \[2.0\times {{10}^{2}}=200.0.\] Now, we will work on our problem. We have \[8\times {{10}^{3}}\] so firstly we can write it as \[8.0\times {{10}^{3}}.\] Now power is positive 3. So, the decimal moves 3 units to the right. So, we get,
\[8.0\times {{10}^{3}}=8000.0\]
\[\Rightarrow 8.0\times {{10}^{3}}=8000\]

Note: Another way to solve is to use Algebra where we will expand the exponent and then simplify using multiplication. That is for example \[2\times {{3}^{3}}=2\times 3\times 3.\] Now, solving this, we get, 18. So in \[8\times {{10}^{3}},\] we will expand \[{{10}^{3}}\] as \[10\times 10\times 10.\] So, we get, \[8\times {{10}^{3}}=10\times 10\times 10.\] On simplifying, we get the answer as 8000. Hence \[8\times {{10}^{3}}\] is the same as 8000.