Question

Write the following numbers in standard form: 0.000000564

Answer 1

Hint: In this question, we have to write 0.000000564 in standard form. For this, we will first understand the format of the standard form and then write the given number in standard form. We will also use properties of exponents given by,
\[\begin{align}
  & \left( i \right){{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}} \\
 & \left( ii \right)\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}} \\
\end{align}\]

Complete step by step answer:
Here, we are given the number as 0.000000564 and we have to write it in standard form. First let us understand the format of standard form.
Any number that we can write as a decimal number between 1.0 and 10.0 multiplied by a power of 10 is said to be in standard form. For number like 0.000000564 standard form is written in following form:
(I) Write the first non-zero digit. Here, we have the first non-zero digit as 5.
(II) Now, add a decimal point after it, so the number becomes 5.
(III) As we can see the decimal point shifts 7 places to the right. So, we will use the power of ten as -7.
(IV) Hence, our number in standard form becomes $5.64\times {{10}^{-7}}$.

Hence, $5.64\times {{10}^{-7}}$ is our required answer.

Note: Students should note that, while converting numbers to standard form, take care in counting the number of digits. When decimal is shifted to right, we use negative sign in the power of ten and when decimal is shifted to left, we use positive sign in the power of ten. We can solve it in following form also, given number is: 0.000000564
It can be written as: $\dfrac{564}{1000000000}=\dfrac{564}{{{10}^{9}}}$.
It can also be written as: $\dfrac{564.00}{{{10}^{9}}}$.
Shifting two points to the left, we get: $\dfrac{5.64\times {{10}^{2}}}{{{10}^{9}}}$.
Now, we know $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ we get: $5.64\times {{10}^{2-9}}=5.64\times {{10}^{-7}}$.
Hence, this is our required answer.