Question

Challenge Problem Find the product and write the answer in scientific notation. $$ \left(5.24 \times 10^{6}\right)\left(6.5 \times 10^{13}\right) $$

Short answer

3.406 \times 10^{20}

Step-by-step solution

Multiply the Coefficients

First, multiply the coefficients of the given numbers. This means multiplying 5.24 by 6.5:\( 5.24 \times 6.5 = 34.06 \)

Add the Exponents

Next, add the exponents of the powers of 10. This involves adding the exponents 6 and 13:\( 10^6 \times 10^{13} = 10^{6+13} = 10^{19} \)

Combine the Results

Combine the result from multiplying the coefficients and the result from adding the exponents:\( 34.06 \times 10^{19} \)

Convert to Scientific Notation

Finally, ensure the result is in proper scientific notation. Since 34.06 is not between 1 and 10, adjust it by moving the decimal point one place to the left, and increase the exponent by 1:\( 34.06 \times 10^{19} = 3.406 \times 10^{20} \)

Key concepts

Multiplying Coefficients

When working with scientific notation, the first step is to multiply the coefficients of the numbers. The coefficient is the digit part of the number in scientific notation, meaning the number before the exponent. Let's use a specific example to make this concept clearer. In the problem, we are given \(5.24 \times 10^6\text{ and }6.5 \times 10^{13} \). The coefficients here are 5.24 and 6.5.

To multiply these coefficients, simply perform a regular multiplication: \(5.24 \times 6.5 = 34.06\). This step is purely arithmetic. You temporarily set aside the powers of 10 and focus solely on these numerical values.

Adding Exponents

Once the coefficients are multiplied, the next part involves handling the exponents. The exponents are the powers of 10 in scientific notation. In this problem, the exponents are 6 and 13 from \(10^6 \text{ and } 10^{13} \).

Adding exponents means you simply add these two numbers together. This is because when you multiply powers of 10, you add their exponents: \(10^6 \times 10^{13} = 10^{6+13} = 10^{19}\). Remember, this rule applies only for multiplication of like bases (in this case, the base is 10). Adding the exponents is a straightforward step but crucial to ensure the accuracy of your final answer.

Converting to Proper Scientific Notation

The last step in solving a problem involving scientific notation is to convert the result to proper scientific notation. The proper form is when the coefficient is a number between 1 and 10. In our example, after multiplying the coefficients (34.06) and adding the exponents (19), our combined result was \(34.06 \times 10^{19}\).

Since 34.06 is not between 1 and 10, we have to adjust it. We do this by moving the decimal point one place to the left, which changes 34.06 to 3.406. Every time we move the decimal point to the left, we add 1 to the exponent of 10:

\34.06 \times 10^{19} = 3.406 \times 10^{20}\

Now, the number is in proper scientific notation, with the coefficient (3.406) falling between 1 and 10 and the exponent (20) adjusted accordingly. This adjustment is essential for representing the number correctly in scientific notation.