Perform the following operation and express the answer in correct scientific notation.
[tex]5.0 \times 10^4 \times 2.9 \times 10^8[/tex]
Enter the coefficient and the exponent.
Answer (2)
Multiply the coefficients: 5.0
\[ \times \]
2.9 = 14.5 .
Multiply the powers of 10: 10^4
\[ \times \]
10^8 = 10^{12} .
Convert the coefficient to scientific notation: 14.5 = 1.45
\[ \times \]
10^1 .
Combine the terms and simplify: 1.45
\[ \times \]
10^1
\[ \times \]
10^{12} = 1.45
\[ \times \]
10^{13} . The final answer is 1.45 × 1 0 13 .
Explanation
Understanding the problem We are given the expression 5.0
\[ \times \]
10^4
\[ \times \]
2.9
\[ \times \]
10^8 . We need to perform the multiplication and express the result in scientific notation. Scientific notation requires the form a
\[ \times \]
10^b where 1 ≤ a < 10 and b is an integer.
Multiplying the coefficients First, we multiply the coefficients: 5.0
\[ \times \]
2.9 = 14.5 .
Multiplying the powers of 10 Next, we multiply the powers of 10: 10^4
\[ \times \]
10^8 = 10^{4+8} = 10^{12} .
Combining the results Now, we combine the results: 14.5
\[ \times \]
10^{12} . However, 14.5 is not in the correct scientific notation because it is greater than 10.
Expressing the coefficient in scientific notation To express 14.5 in scientific notation, we write it as 1.45
\[ \times \]
10^1 .
Combining the terms Finally, we combine the terms: 1.45
\[ \times \]
10^1
\[ \times \]
10^{12} = 1.45
\[ \times \]
10^{1+12} = 1.45
\[ \times\]
10^{13} .
Final answer Therefore, the final answer in scientific notation is 1.45
\[ \times \]
10^{13} .
Examples
Scientific notation is extremely useful in various fields like physics, astronomy, and engineering. For example, in astronomy, the distance to a galaxy might be 3.09 x 10^22 meters. In chemistry, Avogadro's number, which represents the number of atoms or molecules in a mole, is approximately 6.022 x 10^23. Using scientific notation makes it easier to work with very large or very small numbers, simplifying calculations and making them more manageable.
The final result of multiplying 5.0 × 1 0 4 by 2.9 × 1 0 8 in scientific notation is 1.45 × 1 0 13 . We first multiply the coefficients and powers of 10, then express the coefficient in proper scientific notation. Combining these gives us the final answer in scientific notation.
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