Question

Multiply without using a calculator. Give your answer in scientific notation.$$\left(3.0 \times 10^{3}\right)\left(5.0 \times 10^{2}\right)$$

Short answer

1.5 x 10^4

Step-by-step solution

Multiply the coefficients

Multiply the numbers in front of the scientific notation, which are the coefficients, without considering the powers of 10. In this case, multiply 3.0 by 5.0 to get 15.0.

Add the exponents

When multiplying powers of the same base, you add the exponents. Here, add the exponents 3 and 2 from the powers of 10 to get 3 + 2 = 5.

Combine the results in scientific notation

Combine the result from Step 1 and Step 2 to express it in scientific notation. The final answer is 15.0 multiplied by 10 raised to the power of 5, which is written as 1.5 x 10^4 after adjusting the coefficient to one significant decimal.

Key concepts

Multiplying Coefficients

When you multiply numbers in scientific notation, the first step is to multiply the coefficients. These are the numbers that appear before the exponent part. In scientific notation, these coefficients are usually between 1 and 10. It's essential to get this step right because it lays the foundation for the next steps. For example, when you have \(3.0 \times 10^{3}\) and \(5.0 \times 10^{2}\), you start by multiplying the 3.0 by 5.0, which gives you 15.0.

This multiplication is straightforward because you treat the coefficients like any other numbers that you would multiply. If either coefficient had more digits, you'd use the same multiplication methods taught for any numbers. However, since we're working with scientific notation, after multiplying coefficients, additional steps ensure the final number is correctly formatted in scientific notation.

Adding Exponents

Once the coefficients have been multiplied, the next step is to add exponents. In scientific notation, when you multiply two powers of 10, you add the exponents because you're essentially combining the number of times 10 is multiplied by itself.

For instance, \(10^{3}\) means 10 is multiplied by itself 3 times (10 x 10 x 10), and \(10^{2}\) means 10 is multiplied by itself twice (10 x 10). When you multiply them together (\(10^{3} \times 10^{2}\)), you are essentially multiplying ten by itself 3+2, or 5 times, which gives you \(10^{5}\). This rule helps simplify multiplication and ensure that you do the arithmetic correctly while working in scientific notation.

Expressing in Scientific Notation

After multiplying the coefficients and adding the exponents, the final step is to express the product in proper scientific notation. Scientific notation requires the coefficient to be a number between 1 and 10, which sometimes means adjusting the coefficient and modifying the exponent accordingly.

In our example, the coefficient we initially got was 15.0, which is not between 1 and 10. To adjust it, we convert 15.0 to 1.5, and since we moved the decimal point one place to the left, we compensate by increasing the power of 10 by one, turning \(15.0 \times 10^{5}\) into \(1.5 \times 10^{6}\). This expression now correctly follows the scientific notation rules, with the coefficient between 1 and 10 and the exponent representing the order of magnitude. Properly expressing numbers in scientific notation is crucial for maintaining precision and simplicity when working with very large or small numbers.