Question

divide. write your answer in scientific notation. \\(\frac{2.88 \times 10^{-9}}{7.2 \times 10^{-1}}\\)

Explanation:

Step1: Divide the coefficients

Divide \(2.88\) by \(7.2\). So, \(\frac{2.88}{7.2} = 0.4\).

Step2: Divide the powers of 10

Use the rule \( \frac{10^a}{10^b}=10^{a - b}\). Here, \(a=-9\) and \(b = - 1\), so \(10^{-9-(-1)}=10^{-9 + 1}=10^{-8}\).

Step3: Combine the results

Multiply the coefficient from Step 1 and the power of 10 from Step 2. But \(0.4\) in scientific notation is \(4\times10^{-1}\). So, \(4\times10^{-1}\times10^{-8}=4\times10^{-9}\)? Wait, no, wait. Wait, \(0.4\times10^{-8}\). Wait, no, let's redo Step 1. Wait, \(2.88\div7.2\): \(7.2\times0.4 = 2.88\), correct. Then \(10^{-9}\div10^{-1}=10^{-9+1}=10^{-8}\). So then \(0.4\times10^{-8}\). But scientific notation requires the coefficient to be between 1 and 10. So we rewrite \(0.4\) as \(4\times10^{-1}\). Then \(4\times10^{-1}\times10^{-8}=4\times10^{-9}\)? Wait, no, \( - 1+(-8)=-9\)? Wait, no, \(10^{-1}\times10^{-8}=10^{-9}\), yes. Wait, but let's check again. \(2.88\div7.2 = 0.4\). \(10^{-9}\div10^{-1}=10^{-8}\). So \(0.4\times10^{-8}\). To convert \(0.4\) to scientific notation, it's \(4\times10^{-1}\). So \(4\times10^{-1}\times10^{-8}=4\times10^{-9}\)? Wait, no, that can't be. Wait, no, \(0.4\times10^{-8}\) is equal to \(4\times10^{-1}\times10^{-8}=4\times10^{-9}\)? Wait, let's calculate the original division: \(\frac{2.88\times10^{-9}}{7.2\times10^{-1}}=\frac{2.88}{7.2}\times\frac{10^{-9}}{10^{-1}} = 0.4\times10^{-8}\). Now, \(0.4\times10^{-8}\) in proper scientific notation (coefficient between 1 and 10) is \(4\times10^{-1}\times10^{-8}=4\times10^{-9}\)? Wait, no, \(0.4\times10^{-8}=4\times10^{-1}\times10^{-8}=4\times10^{-9}\). Wait, but let's do the division directly: \(2.88\div7.2 = 0.4\), \(10^{-9}\div10^{-1}=10^{-8}\), so \(0.4\times10^{-8}\). But \(0.4\) is \(4\times10^{-1}\), so \(4\times10^{-1}\times10^{-8}=4\times10^{-9}\). Wait, but let's check with a calculator: \(2.88\times10^{-9}=0.00000000288\), \(7.2\times10^{-1}=0.72\). Divide them: \(0.00000000288\div0.72 = 0.000000004 = 4\times10^{-9}\). Yes, that's correct.

Wait, but maybe I made a mistake in the exponent. Let's recall the rule for dividing exponents: \( \frac{10^m}{10^n}=10^{m - n}\). So here, \(m=-9\), \(n=-1\), so \(m - n=-9-(-1)=-8\). So the exponent part is \(10^{-8}\). Then the coefficient is \(2.88\div7.2 = 0.4\). So we have \(0.4\times10^{-8}\). Now, to write this in scientific notation, the coefficient must be between 1 and 10. So we move the decimal point in \(0.4\) one place to the right, which gives \(4\), and we decrease the exponent by 1 (because we multiplied by \(10^1\) to move the decimal, so we have to divide by \(10^1\) to keep the value the same). So \(0.4\times10^{-8}=4\times10^{-1}\times10^{-8}=4\times10^{-9}\). Yes, that's correct.

Answer:

\(4\times10^{-9}\)