Problem 64 - Elementary Algebra

Question

Perform the following operations. $$ \left(6 \times 10^{14}\right)\left(6 \times 10^{-10}\right) $$

Step-by-Step Solution

Answer

Question: Multiply the given numbers in scientific notation: $$ (6 \times 10^{14})(6 \times 10^{-10}) $$ Answer: The result of the given multiplication is: $$ (6 \times 10^{14})(6 \times 10^{-10}) = 3.6 \times 10^5 $$

Step 1: Multiply the coefficients

Multiply the coefficients (6 and 6) to get the new coefficient. $$ 6 \times 6 = 36 $$

Step 2: Add the exponents

Add the exponents (14 and -10) to get the new exponent. $$ 14 + (-10) = 4 $$

Step 3: Combine the new coefficient and exponent

Combine the new coefficient (36) and the new exponent (4) in scientific notation. $$ 36 \times 10^4 $$

Step 4: Check if adjustment is needed for scientific notation

The coefficient should be between 1 and 10. Since our coefficient is 36, which is greater than 10, we need to adjust it. Divide 36 by 10 to get 3.6. $$ 3.6 \times 10^5 $$ So, the result of the given multiplication is: $$ \left(6 \times 10^{14}\right)\left(6 \times 10^{-10}\right) = 3.6 \times 10^5 $$

Key Concepts

Multiplying Powers of Ten

When we multiply numbers in scientific notation, we often deal with powers of ten. These powers of ten allow us to express very large or small numbers compactly. To multiply powers of ten, you simply add their exponents. Here's why this works: when you multiply like bases, you keep the base and add the exponents. For example, when multiplying \(10^{14}\) by \(10^{-10}\), you add the exponents:

• \(14 + (-10) = 4\)

Thus, the resulting power of ten in our example becomes \(10^4\). By adding exponents, you ensure the number remains in scientific notation, making it both easier to handle and understand.

Coefficient Multiplication

Coefficients are the numbers that are multiplied by the power of ten in scientific notation. They are usually numbers between 1 and 10, but sometimes they may not be, especially after initial calculations. In our example, both coefficients are 6. Multiplying them is straightforward:

• \(6 \times 6 = 36\)

This step is crucial because the coefficients set the magnitude of your number. Once you multiply the coefficients, you combine the result with the power of ten obtained earlier, setting up your final answer in scientific form.

Exponent Rules

Exponent rules are like shortcuts that save time and effort in math calculations. The key rule for multiplying numbers in scientific notation is to add the exponents. This comes from the law of exponents:

• When you multiply terms with the same base, such as \(10^a\) and \(10^b\), you simply add the exponents (\(a + b\)).

This principle helps maintain structure and simplifies calculations significantly. Whether the exponents are positive or negative, this rule holds true. So, when you encounter multiplication like \(10^{14}\) and \(10^{-10}\), just remember to perform basic addition on the exponents for an efficient solution.

Adjusting Scientific Notation

Sometimes, when multiplying coefficients, you'll end up with a number that doesn't fit the standard form of scientific notation — which is a number between 1 and 10. Here, our calculated coefficient was 36, which is too large. To adjust it, we divide by 10 to shift the decimal place:

• \(36 \div 10 = 3.6\)

By doing so, we balance the scale by adjusting the exponent of our power of ten accordingly, thereby increasing it by one:

• The number becomes \(3.6 \times 10^5\) since one exponent is added.

This adjustment keeps our final expression true to the correct form of scientific notation, facilitating an accurate and readable result.