Question

think about the process the length of cell a is 6×10⁻⁵ m. the length of cell b is 0.000003 m. what is the ratio of cell a’s length to cell b’s length? use pencil and paper. is it easier to find the ratio when the numbers are expressed in scientific notation or in standard form? explain your reasoning. the ratio of cell a’s length to cell b’s length is \boxed{}. (type the ratio as a simplified fraction.)

Explanation:

Step1: Convert cell B's length to scientific notation

Cell B's length is \(0.000003\) m. In scientific notation, this is \(3\times10^{-6}\) m.

Step2: Find the ratio of cell A's length to cell B's length

Cell A's length is \(6\times10^{-5}\) m. The ratio is \(\frac{6\times10^{-5}}{3\times10^{-6}}\). Using the rule of exponents \(\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}\), we have \(\frac{6}{3}\times10^{-5-(-6)} = 2\times10^{1}=20\)? Wait, no, wait. Wait, we need a fraction. Wait, let's do it as fractions. \(6\times10^{-5}=\frac{6}{10^{5}}\), \(3\times10^{-6}=\frac{3}{10^{6}}\). Then the ratio is \(\frac{\frac{6}{10^{5}}}{\frac{3}{10^{6}}}=\frac{6}{10^{5}}\times\frac{10^{6}}{3}=\frac{6\times10^{6}}{3\times10^{5}}=\frac{6}{3}\times10^{6 - 5}=2\times10 = 20\)? But the problem says to type as a simplified fraction. Wait, maybe I made a mistake. Wait, \(6\times10^{-5}\) is \(0.00006\), and \(0.000003\) is cell B. So \(0.00006\div0.000003 = 20\), which is \(\frac{20}{1}\) or just 20. Wait, but let's check the scientific notation approach again. \(\frac{6\times10^{-5}}{3\times10^{-6}}=\frac{6}{3}\times10^{-5 + 6}=2\times10^{1}=20\), which is \(\frac{20}{1}\). So the simplified fraction is 20 (or \(\frac{20}{1}\)).

Wait, maybe I messed up the exponent subtraction. Wait, when dividing exponents with the same base, \(a^{m}\div a^{n}=a^{m - n}\). So \(10^{-5}\div10^{-6}=10^{-5-(-6)}=10^{1}\). Then \(\frac{6}{3}=2\), so \(2\times10^{1}=20\), which is \(\frac{20}{1}\).

Answer:

\(\boxed{20}\) (or \(\boxed{\frac{20}{1}}\))