Convert the following number into correct scientific notation.
[tex]
\begin{array}{l}
56.42 \times 10^{-6} \
{[?] \times 10^{[?]}}
\end{array}
[/tex]
Enter the coefficient in the green box and the exponent in the yellow box.
Coefficient $\square$
Exponent $\square$
Answer (2)
The number 56.42 × 1 0 − 6 in proper scientific notation is 5.642 × 1 0 − 5 . The coefficient is 5.642 and the exponent is − 5 .
;
Divide the coefficient 56.42 by 10 to get a value between 1 and 10: 5.642 .
Compensate by multiplying the power of 10 by 10: 1 0 − 6 × 1 0 1 .
Simplify the expression using exponent rules: 5.642 × 1 0 1 + ( − 6 ) = 5.642 × 1 0 − 5 .
The number in scientific notation is 5.642 × 1 0 − 5 .
Explanation
Understanding Scientific Notation We are given the number $56.42
\times 10^{-6} an d a s k e d t oco n v er t i tt osc i e n t i f i c n o t a t i o n . S c i e n t i f i c n o t a t i o n re q u i res t h e n u mb er t o b e in t h e f or m a
\times 10^{b}$, where 1 ≤ ∣ a ∣ < 10 and b is an integer.
Adjusting the Coefficient The coefficient 56.42 is not between 1 and 10, so we need to adjust it. To do this, we can divide 56.42 by 10 to get 5.642 , which is between 1 and 10.
Compensating for the Change Since we divided the coefficient by 10, we need to multiply by 10 to compensate. So we have $56.42
\times 10^{-6} = 5.642
\times 10
\times 10^{-6}$.
Simplifying the Exponent Now we can use the exponent rule $10^{a}
\times 10^{b} = 10^{a+b}$ to simplify the expression: $5.642
\times 10^{1}
\times 10^{-6} = 5.642
\times 10^{1 + (-6)} = 5.642
\times 10^{-5}$.
Final Answer Therefore, the number in scientific notation is $5.642
\times 10^{-5}$. The coefficient is 5.642 and the exponent is − 5 .
Examples
Scientific notation is used in many fields, such as physics, astronomy, and chemistry, to represent very large or very small numbers. For example, the speed of light is approximately 3 × 1 0 8 meters per second, and the mass of an electron is approximately 9.11 × 1 0 − 31 kilograms. Using scientific notation makes it easier to work with these numbers and compare their magnitudes. It's also used in computer science to represent floating-point numbers.
