Question

How do you write $9.3\times {{10}^{-5}}$ in the standard notation?

Answer 1

Hint: We are given a term as $9.3\times {{10}^{-5}}$ we are asked to write it in standard notation, to do so we will learn what type of notation is this in which it is written then we learn about standard notation we will use the property that ${{x}^{a}}=x\times x\times x\times .......\times x$ to simplify and solve using this knowledge we be able to change $8\times 10$ into standard notation.

Complete step by step solution:
We are given a term as $9.3\times {{10}^{-5}}$ if we look closely, we see that term has one term as $9.3$ and other is exponent of 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 standard notation mean to unite the term back to the decimal form to do so we have to understand that how the scientific notation were decimal if we have decimal term is when non-zero term is after $3$ unit from the decimal to the right then the power of exponent will run to negative of $3$ for example $0.003$ it will be come and if the non-zero term is after say $3$ unit from the decimal to the left that the power will remain to the $+3$ for example $3000.0$ it will become $3\times {{10}^{3}}$. Similarly, we just had to use this knowledge in reverse order. That is power of $10$positive then decimal will move to the right and if the power is negative then decimal will move to the left. For example, $2.0\times {{10}^{2}}$ power is positive $2$ so decimal will move $2$ units to right so, $2.0\times {{10}^{2}}=200.0$
Now we work on our problem we have $9.3\times {{10}^{-5}}$ so firstly we will observe that power of is negative its power is $10$ so the decimal will move to the left, its power is $-5$, so it will move $5$ unit to the left.
In $9.3$ there is just one unit to the left so to fulfil other $4$, we will put zero so $9.3\times {{10}^{-5}}=0.000093$ hence our problem $9.3\times {{10}^{-5}}$in standard notation is given as $0.000093$

Note: Other way to solve is to use the algebra where we will expand the exponent and then simplify why multiplication that is for example $2\times {{3}^{3}}=2\times 3\times 3$ now solving we get $18$. So, in $8\times {{10}^{3}}$, we will expand ${{10}^{3}}$ as $10\times 10\times 10$. So, we get $8\times {{10}^{3}}=8\times 10\times 10\times 10$. Simplifying we get $8000$
Hence $9.3\times {{10}^{-5}}$ is the same as $8000$. So in $9.3\times {{10}^{-5}}$ as we know that ${{a}^{-b}}=\dfrac{1}{{{a}^{b}}}$ so $9.3\times {{10}^{-5}}=\dfrac{9.3}{{{10}^{5}}}$
Now as ${{10}^{5}}=10\times 10\times 10\times 10\times 10=100000$
So we divide $9.3$ by $100000$
$\Rightarrow 9.3\times {{10}^{-5}}=0.000093$