Problem 89 - College Algebra

Question

Perform the indicated computations. Write the answers in scientifi c notation. If necessary, round the decimal factor in your scientifi c notation answer to two decimal places. $$\left(1.6 \times 10^{15}\right)\left(4 \times 10^{-11}\right)$$

Step-by-Step Solution

Answer

The result is \(6.4 \times 10^{4}\)

Step 1: Multiply Decimal Parts

We start by multiplying the decimal parts together, that is, \(1.6 \times 4 = 6.4\).

Step 2: Add Exponents

Secondly, we add the exponents, this means \(10^{15} \times 10^{-11} = 10^{15+(-11)} = 10^{4}\).

Step 3: Combine and Present in Scientific Notation

Finally, we combine the results of step 1 and step 2 to obtain \(6.4 \times 10^{4}\). This is already in scientific notation and does not need to be rounded as it already has only one decimal place.

Key Concepts

Multiplying Decimals

Multiplying decimals is a crucial step when working with scientific notation. In scientific problems, you often deal with numbers that have decimal points, and knowing how to multiply them accurately is important.
To multiply decimals:

• Ignore the decimal points initially and multiply the numbers as if they were whole numbers.
• After obtaining the product, count the total number of decimal places in both the numbers you multiplied.
• Insert the decimal point in your product to have the same number of decimal places.

For example, multiplying 1.6 by 4 requires you to first consider 16 and 4, which gives 64. Since there is one decimal place in 1.6, place the decimal point to make 6.4. This simple process ensures precision in computations, especially in scientific contexts.

Exponent Addition

Exponent addition is an integral part of manipulating expressions in scientific notation. When you multiply numbers in scientific notation, you specifically add their exponents if the bases are the same.
Here’s how it works:

• Ensure that the base for both numbers is the same, typically base 10 in scientific notation problems.
• Add the exponents together directly.

For instance, if you have \[10^{15} \times 10^{-11}\],you add the exponents: 15 and -11. This results in \[10^{4}\].This simple addition highlights the power of working with exponents — it simplifies multiplying large or tiny numbers, making them more manageable.

Powers of Ten

The concept of powers of ten is at the core of scientific notation. It's what allows us to handle very large or very small numbers efficiently. Scientific notation represents numbers as a product of a decimal and a power of ten, making calculations simpler.
Here's a breakdown:

• A positive exponent (e.g., \[10^{4}\]) means multiply the decimal by 10 raised to that power, making the number larger.
• A negative exponent (e.g., \[10^{-11}\]) means divide the decimal, effectively creating a very small number.

This method helps unify the approach to handling numbers of all sizes. By manipulating the exponent, you transform numbers into more understandable formats, especially useful in scientific contexts. Understanding powers of ten enables you to improve your proficiency in scientific notation and similar mathematical concepts.