Problem 91 - Algebra and Trigonometry

Question

Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$ \left(6.1 \times 10^{-8}\right)\left(2 \times 10^{-4}\right) $$

Step-by-Step Solution

Answer

The scientific notation for the multiplication of the given numbers is \(1.22 \times 10^{-11}\).

Step 1: Multiply the Numbers

Multiply the two numbers without the powers of 10. That is, multiply \(6.1\) by \(2\) which gives \(12.2\).

Step 2: Add the powers

Add the powers separately. That's \( -8 + (-4)\), which equals to \(-12\).

Step 3: Combine Results

Combine the two results creating a number in scientific notation. This gives \(12.2 \times 10^{-12}\).

Step 4: Adjust to Proper Scientific Notation

The number should have only one digit before the decimal point in scientific notation. Dividing the number part by \(10\) and adding \(1\) to the exponent will give the number in correct scientific notation format. So, \(12.2\) becomes \(1.22\) and \(-12\) becomes \(-11\), making the final answer \(1.22 \times 10^{-11}\).

Key Concepts

Understanding Exponents

Exponents play a crucial role in helping us simplify and express large or very tiny numbers through multiplication. When we have a number such as \(10^3\), it means we multiply 10 by itself three times, resulting in 1000. The exponent, in this case 3, tells us how many times the base (10) will be used as a factor.

Exponents can also be negative, such as in \(10^{-8}\). This indicates division or the reciprocal of the base raised to the positive exponent. So, \(10^{-8}\) means \(\frac{1}{10^8}\). This makes the number very small, as we're essentially dividing 1 by a very large number.

The ability to manipulate exponents is fundamental in scientific notation, where these powers of ten allow us to clearly and efficiently indicate the scale of a number.

Multiplication of Numbers in Scientific Notation

Multiplying numbers in scientific notation is straightforward once you break it down. Here’s the step-by-step breakdown:

• First, ignore the powers of ten. Multiply the non-exponential numbers. If given \(6.1\) and \(2\), just calculate \(6.1 \times 2\), which equals \(12.2\).
• Next, handle the exponents separately by adding them together. Take the powers of ten from the two numbers and sum them. For example, \(10^{-8}\) and \(10^{-4}\) become \(10^{-12}\) when added.

Rejoining these two results gives a new number in scientific notation form, like \(12.2 \times 10^{-12}\). It's essential to keep the decimal factor (here \(12.2\)) separate initially, so these operations remain clear and organized.

Rounding to Two Decimal Places

Decimal places require us to round to a specific number of digits after the decimal point. In scientific notation, if necessary, we might need to adjust our result to two decimal places.

After initial multiplication, you could end with a vertical bar result like \(12.2\). However, when transforming it into correct scientific notation, conventions demand only one numeral before the decimal. Here, \(12.2\) needs continuation to \(1.22 \times 10^{-11}\).

If the original calculation provides more lengthy decimals, like \(12.239:\), you would round this to two decimal places for simplicity and cleaner presentation. Thus, \(12.239\) becomes \(12.24\) when rounded.

Powers of Ten in Scientific Notation

Powers of ten are the backbone of scientific notation, allowing us to represent numbers that are too large or too small using a compact form. A power of ten is expressed as \(10^n\), where \(n\) is an integer.

In scientific notation, any number is written as the product of a number between 1 and 10, and a power of ten. For instance, \(1.22 \times 10^{-11}\) uses \(10^{-11}\) to define the scale, indicating it's a very small value.

Understanding how these powers work allows you to easily read and write numbers regardless of their size. It's the ideal method to succinctly communicate magnitude, especially in science and engineering fields, where these notations save space and time.