Question

write explain why anyone would want to use scientific notation.
multiple choice what is (3.4 × 10¹⁰)(4.8 × 10⁵) in scientific notation?

Explanation:

First Part (Explanation for Using Scientific Notation)

Scientific notation is used to express very large or very small numbers in a concise and manageable form. It simplifies calculations involving such numbers, makes comparisons easier, and is a standard way to represent numbers in scientific and technical fields (e.g., astronomy, microbiology) where extreme magnitudes are common. For example, writing the distance to the sun as \(1.496\times10^{11}\) meters is clearer than writing out all the digits.

Step 1: Multiply the coefficients and the powers of 10 separately.

For the coefficients: \(3.4\times4.8 = 16.32\)
For the powers of 10: Using the rule \(a^m\times a^n=a^{m + n}\), we have \(10^{10}\times10^{5}=10^{10 + 5}=10^{15}\)

Step 2: Combine the results and adjust to scientific notation.

We get \(16.32\times10^{15}\), but scientific notation requires the coefficient to be between 1 and 10. So we rewrite \(16.32\) as \(1.632\times10^{1}\). Then multiply by \(10^{15}\): \(1.632\times10^{1}\times10^{15}=1.632\times10^{16}\) (assuming the original numbers are \(3.4\times10^{10}\) and \(4.8\times10^{5}\); if there was a typo and the first exponent was different, the process is similar: multiply coefficients, add exponents, adjust coefficient to 1 - 10).

Answer:

Scientific notation is used to concisely represent very large or very small numbers, simplify calculations/comparisons, and is standard in scientific/technical fields.

Second Part (Calculating \((3.4\times10^{10})(4.8\times10^{5})\) in Scientific Notation)