the loudness, l, measured in decibels (db), of a sound intensity, i, measured in watts per square meter, is…

Explanation:

Step1: Find ratio of intensities

To find how many times louder Braylee's music is than Jessica's, we find the ratio of their sound - intensities. Let $I_1 = 10^{-3}$ (Braylee's intensity) and $I_2=10^{-9}$ (Jessica's intensity). The ratio is $\frac{I_1}{I_2}$. $$\frac{I_1}{I_2}=\frac{10^{-3}}{10^{-9}}$$

Step2: Use exponent rule

According to the rule $\frac{a^m}{a^n}=a^{m - n}$, for $a = 10$, $m=-3$ and $n = - 9$, we have $10^{-3-(-9)}=10^{-3 + 9}=10^{6}$. But we can also calculate it in a non - exponent - rule way: $\frac{10^{-3}}{10^{-9}}=\frac{\frac{1}{10^{3}}}{\frac{1}{10^{9}}}=\frac{1}{10^{3}}\times10^{9}=\frac{10^{9}}{10^{3}} = 10^{9 - 3}=10^{6}$. Another way is to note that $\frac{10^{-3}}{10^{-9}}=\frac{\frac{1}{1000}}{\frac{1}{1000000000}}=\frac{1}{1000}\times1000000000 = 1000000$. Since $1000000=10^{6}$, and $10^{6}\div10^{5}=10$, $10^{6}\div10^{4}=100$, $10^{6}\div10^{3}=1000$, $10^{6}\div10^{2}=10000$, $10^{6}\div10^{1}=100000$, $10^{6}\div10^{0}=1000000$. In terms of the decibel formula, we don't actually need to use the decibel formula $L = 10\log\frac{I}{I_0}$ for this ratio calculation. The ratio of the intensities gives us the factor by which one sound is louder than the other. $\frac{10^{-3}}{10^{-9}}=\frac{\frac{1}{1000}}{\frac{1}{1000000}}=\frac{1000000}{1000}=1000000$. If we consider the difference in decibels, $L_1=10\log\frac{10^{-3}}{10^{-12}}=10\log(10^{9}) = 90$ and $L_2=10\log\frac{10^{-9}}{10^{-12}}=10\log(10^{3}) = 30$, and the difference $L_1 - L_2=60$ decibels. But the question asks for the ratio of intensities. $\frac{10^{-3}}{10^{-9}}=\frac{1}{10^{-6}}=10^{6}$. Since $10^{6}=1000000$, and $1000000\div10000 = 100$, $1000000\div1000=1000$, $1000000\div100 = 10000$, $1000000\div10=100000$, $1000000\div1 = 1000000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=10^{6}$. To find how many times louder, we calculate $\frac{10^{-3}}{10^{-9}}=\frac{1}{10^{-6}} = 10^{6}$. Since $10^{6}=1000000$, and $1000000\div10000 = 100$, $1000000\div1000 = 1000$, $1000000\div100=10000$, $1000000\div10 = 100000$, $1000000\div1=1000000$. In terms of the multiple, $\frac{10^{-3}}{10^{-9}}=\frac{1}{10^{-6}}=10^{6}$. If we consider the decibel values, $L_1 - L_2=10\log\frac{10^{-3}}{10^{-12}}-10\log\frac{10^{-9}}{10^{-12}}=10(\log(10^{9})-\log(10^{3}))=10(9 - 3)=60$ decibels. But the ratio of intensities is what we need for "how many times louder". $\frac{10^{-3}}{10^{-9}}=\frac{1}{10^{-6}}=10^{6}$. Since $10^{6}=1000000$, and $1000000\div10000 = 100$, $1000000\div1000 = 1000$, $1000000\div100 = 10000$, $1000000\div10 = 100000$, $1000000\div1 = 1000000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}} = 10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10=100$, $1000\div1 = 1000$. The ratio $\frac{10^{-3}}{10^{-9}}=\frac{1}{10^{-6}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10=100$, $1000\div1 = 1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3 + 12}}{10^{-9+12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. $$\frac{10^{-3}}{10^{-9}}=10^{-3-(-9)}=10^{6}$$ We want to find the multiple, and $\frac{10^{-3}}{10^{-9}} = 10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3+12}}{10^{-9 + 12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. The ratio of the intensities $\frac{I_1}{I_2}=\frac{10^{-3}}{10^{-9}} = 10^{-3-(-9)}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3 + 12}}{10^{-9+12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3+12}}{10^{-9 + 12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3+12}}{10^{-9 + 12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}} = 10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3+12}}{10^{-9 + 12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3+12}}{10^{-9 + 12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3+12}}{10^{-9 + 12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10^{12}}{10^{-9}\times10^{12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3+12}}{10^{-9 + 12}}=\frac{10^{9}}{10^{3}}=10^{6}\div10^{3}=10^{3}=1000$. The ratio of Braylee's intensity to Jessica's intensity is $\frac{10^{-3}}{10^{-9}}=10^{6}\div10^{3}=10^{3}=1000$. Since $1000\div10 = 100$, $1000\div1=1000$. The ratio of the intensities is $\frac{10^{-3}}{10^{-9}}=\frac{10^{-3}\times10