Compute $324,000 \times 30,000^3$ In Scientific Notation

In this article, we will walk through the process of computing the number $324,000 \times 30,000^3$ and express the result in scientific notation. Scientific notation is a way of writing very large or very small numbers in a compact and easily readable format. It is expressed as $a \times 10^b$, where $1 \le |a| < 10$ and $b$ is an integer. This notation is particularly useful in scientific and engineering calculations where dealing with numbers with many digits is common. Our task is to perform the multiplication and convert the result into this standard form. This involves several steps, including converting the original numbers into scientific notation, applying the exponent, performing the multiplication, and finally adjusting the result to fit the scientific notation format. Let's dive into the step-by-step solution to understand each part clearly.

Step 1: Convert Numbers to Scientific Notation

The first step in calculating $324,000 \times 30,000^3$ is to convert the numbers 324,000 and 30,000 into scientific notation. Scientific notation expresses a number as a product of a coefficient (a number between 1 and 10) and a power of 10. This format simplifies calculations and makes it easier to handle very large or very small numbers.

For the number 324,000, we need to move the decimal point five places to the left to get 3.24, which is between 1 and 10. Therefore, we can write 324,000 as $3.24 \times 10^5$. The exponent 5 indicates that we moved the decimal point five places. This conversion is a fundamental step in simplifying our original expression.

Next, we convert 30,000 to scientific notation. Moving the decimal point four places to the left gives us 3. Thus, 30,000 can be written as $3 \times 10^4$. This notation helps us in dealing with large numbers more efficiently. By converting both numbers into scientific notation, we prepare the expression for easier calculation of the power and the subsequent multiplication. The importance of this step lies in its ability to reduce the complexity of the calculation by breaking down large numbers into more manageable components.

Step 2: Apply the Exponent

Now that we have expressed 324,000 and 30,000 in scientific notation, the next step is to apply the exponent to the term $30,000^3$. Recall that 30,000 in scientific notation is $3 \times 10^4$. So, we need to calculate $(3 \times 10^4)^3$.

To do this, we apply the power rule, which states that $(ab)^n = a^n \times b^n$. Applying this rule to our expression, we get $(3 \times 10^4)^3 = 3^3 \times (10^4)^3$. Now, we calculate $3^3$, which is $3 \times 3 \times 3 = 27$. Next, we handle the exponentiation of the power of 10. The rule for this is $(10^a)^b = 10^{a \times b}$. Therefore, $(10^4)^3 = 10^{4 \times 3} = 10^{12}$.

Combining these results, we have $(3 \times 10^4)^3 = 27 \times 10^{12}$. This intermediate result shows how the exponent affects both the coefficient and the power of 10. This step is critical because it significantly increases the magnitude of the number, which is reflected in the large exponent of 10. Simplifying the expression in this way makes it easier to perform the subsequent multiplication.

Step 3: Perform the Multiplication

With $30,000^3$ calculated as $27 \times 10^{12}$, we can now perform the multiplication $324,000 \times 30,000^3$. Recall that we expressed 324,000 in scientific notation as $3.24 \times 10^5$. So, the multiplication becomes $(3.24 \times 10^5) \times (27 \times 10^{12})$.

To multiply these two numbers in scientific notation, we multiply the coefficients and add the exponents of 10. This is based on the properties of exponents, which allow us to simplify the expression. Multiplying the coefficients, we get $3.24 \times 27$. This calculation results in 87.48.

Next, we add the exponents of 10: $10^5 \times 10^{12} = 10^{5 + 12} = 10^{17}$. Combining the results, we have $87.48 \times 10^{17}$. This intermediate result is a large number, but it's not yet in proper scientific notation because 87.48 is not between 1 and 10. The goal of this step is to combine the two scientific notation expressions into a single expression that can be easily converted into standard scientific notation form. Thus, we proceed to the final step to adjust this result.

Step 4: Adjust to Scientific Notation

The final step is to adjust the result, $87.48 \times 10^{17}$, to fit the standard scientific notation form, where the coefficient must be between 1 and 10. Currently, our coefficient is 87.48, which is greater than 10. To correct this, we need to move the decimal point one place to the left, which means dividing by 10. To compensate for this division, we must multiply the power of 10 by 10, effectively increasing the exponent by 1.

Moving the decimal point one place to the left in 87.48 gives us 8.748. So, we have $8.748 \times 10^{17}$. To adjust the exponent, we add 1 to the current exponent of 17, resulting in $10^{17+1} = 10^{18}$. Thus, our final result in scientific notation is $8.748 \times 10^{18}$.

This adjustment ensures that the number is expressed in the correct format for scientific notation, making it easier to compare with other numbers and use in further calculations. The precision of the coefficient (8.748) is maintained, while the magnitude of the number is clearly represented by the power of 10. This final step is crucial for expressing the answer in a standardized and easily understandable form. Therefore, the correct answer is $8.748 \times 10^{18}$.

Final Answer

After performing the calculation and converting the result to scientific notation, we find that $324,000 \times 30,000^3 = 8.748 \times 10^{18}$. This process involved converting the original numbers to scientific notation, applying the exponent, performing the multiplication, and adjusting the result to fit the scientific notation format. Each step is crucial to ensure accuracy and clarity in the final answer. Therefore, the correct answer is:

a. $8.748 \times 10^{18}$