Question

find the product. write the answer in scientific notation
(5.82 × 10³)(6.13 × 10⁴)

Explanation:

Step1: Use commutative and associative properties

We can rearrange the factors using the commutative property of multiplication (\(a\times b = b\times a\)) and associative property of multiplication (\((a\times b)\times c=a\times(b\times c)\)) to group the decimal numbers and the powers of 10 together. So, \((5.82\times 10^{9})(6.13\times 10^{11})=(5.82\times 6.13)\times(10^{9}\times 10^{11})\)

Step2: Multiply the decimal numbers

First, multiply \(5.82\) and \(6.13\). \(5.82\times6.13 = 5.82\times(6 + 0.13)=5.82\times6+5.82\times0.13 = 34.92+0.7566=35.6766\)

Step3: Multiply the powers of 10

When multiplying powers of 10 with the same base, we use the rule \(a^{m}\times a^{n}=a^{m + n}\). So, \(10^{9}\times10^{11}=10^{9 + 11}=10^{20}\)

Step4: Combine the results

Now we combine the two results from Step 2 and Step 3: \(35.6766\times10^{20}\)

Step5: Convert to proper scientific notation

Scientific notation requires that the coefficient is between 1 and 10. So we rewrite \(35.6766\) as \(3.56766\times 10^{1}\). Then we have \(3.56766\times 10^{1}\times10^{20}\). Using the rule \(a^{m}\times a^{n}=a^{m + n}\) again, we get \(3.56766\times10^{1+20}=3.56766\times 10^{21}\)

Answer:

\(3.56766\times 10^{21}\)