Problem 90 - Elementary Algebra

Question

Find the product for the following problems. Write the result in scientific notation. $$ \left(9.3806 \times 10^{52}\right)\left(1.009 \times 10^{-31}\right) $$

Step-by-Step Solution

Answer

Question: Find the product of the following numbers in scientific notation: $(9.3806 \times 10^{52})(1.009 \times 10^{-31})$. Answer: The product of the given numbers in scientific notation is approximately $9.4696 \times 10^{21}$.

Step 1: Recall the rules for multiplying numbers in scientific notation

To multiply numbers in scientific notation, we have to multiply the coefficients and then add the exponents of the powers of 10. In this case, we are given the two numbers as: $$(9.3806 \times 10^{52})(1.009 \times 10^{-31})$$

Step 2: Multiply the coefficients

Multiply the coefficients of the given numbers, which are 9.3806 and 1.009: $$9.3806 \times 1.009 \approx 9.4696$$

Step 3: Add the exponents

Now, we add the exponents in the powers of 10: $$10^{52} \times 10^{-31} = 10^{52-31} = 10^{21}$$

Step 4: Write the result in scientific notation

Finally, to write the product in scientific notation, we combine the coefficient (9.4696) and the power of 10 (10^21): $$\left(9.3806 \times 10^{52}\right)\left(1.009 \times 10^{-31}\right) \approx 9.4696 \times 10^{21}$$

Key Concepts

Multiplying Numbers in Scientific Notation

When you're dealing with numbers in scientific notation, multiplying them is a cinch once you know the simple rule. Scientific notation is a way to express very large or very small numbers succinctly. It involves two parts:

• The coefficient: a number usually between 1 and 10
• A power of ten expressed as \( 10^n \)

To multiply such numbers, follow these steps:

• First, multiply the coefficients. In our case: \( 9.3806 \, \text{and} \, 1.009 \)
• Second, add the exponents of the ten's powers. For example: \( 10^{52} \, \text{and} \, 10^{-31} \)

This method allows you to handle complex numbers by breaking the process into simpler, manageable parts. Once the coefficients are multiplied and the exponents are added, combine them to obtain the product in scientific notation.

Coefficients in Scientific Notation

The coefficient in scientific notation is the number part that's typically between 1 and 10. It's important because it determines the significant digits of the number. When multiplying coefficients, as we did with \( 9.3806 \times 1.009 \), you simply multiply them like regular numbers.

In the exercise, we got:

• \( 9.3806 \times 1.009 \approx 9.4696 \)

It's crucial to ensure the product stays in the appropriate range (1 to 10), so the notation remains valid. Should the product exceed 10, you would need to adjust the coefficient and exponent accordingly. This may involve additional steps like normalizing the coefficient. By keeping the coefficient in the right range, scientific notation remains a powerful tool for simplifying computations and presenting results clearly.

Adding Exponents

Adding exponents is a key part when multiplying numbers in scientific notation. This step utilizes the property of exponents which states that when you multiply like bases, you add their exponents together. Here's how it works:

• Given exponents in the example: \( 10^{52} \) and \( 10^{-31} \)
• Combine them by adding: \( 52 + (-31) = 21 \)

Thus, the power of ten becomes \( 10^{21} \). This is straightforward because it involves simple arithmetic with the exponents. Remember, this property ensures that the operations stay under the laws of exponents, maintaining simplicity and consistency in results.

When you correctly add exponents, it helps to preserve the precision and scale of the number you're working with. This approach keeps the calculations neat and tidy, allowing you to express the final result accurately in scientific notation.