Question

1. a certain star is 4.3 × 10^2 light years from earth. one light year is about 5.9 × 10^12 miles. how far from earth (in miles) is the star? express your answer in scientific notation.

Explanation:

Step1: Recall the multiplication formula for distance

The distance $d$ in miles is the product of the number of light - years $n$ and the number of miles in one light - year $m$. Here, $n = 4.3\times10^{2}$ and $m=5.9\times 10^{12}$. So $d=n\times m=(4.3\times10^{2})\times(5.9\times 10^{12})$.

Step2: Use the commutative and associative properties of multiplication

$(4.3\times10^{2})\times(5.9\times 10^{12})=(4.3\times5.9)\times(10^{2}\times10^{12})$.

Step3: Calculate the product of the non - exponential parts and the exponential parts separately

First, calculate $4.3\times5.9 = 25.37$. Then, use the rule of exponents $a^{m}\times a^{n}=a^{m + n}$, so $10^{2}\times10^{12}=10^{2 + 12}=10^{14}$.

Step4: Convert to scientific notation

Since scientific notation is of the form $a\times10^{n}$ where $1\leqslant a<10$, rewrite $25.37\times10^{14}$ as $2.537\times10^{1}\times10^{14}$. Using the exponent rule again, $2.537\times10^{1}\times10^{14}=2.537\times10^{1 + 14}=2.537\times10^{15}$.

Answer:

$2.537\times 10^{15}$ miles