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Chapter 5: Problem 51

Multiply. Give all answers in scientific notation. See Example 4. $$ \left(9.0 \times 10^{-1}\right)\left(8.0 \times 10^{-6}\right) $$

Short Answer

Expert verified
The product is \( 7.2 \times 10^{-6} \) in scientific notation.

Step by step solution

01

Multiply the Coefficients

First, focus on the coefficients of the numbers in scientific notation. Multiply 9.0 by 8.0. \[ 9.0 \times 8.0 = 72.0 \]
02

Add the Exponents

Next, deal with the powers of ten. Add the exponents from the scientific notation:\[ -1 + (-6) = -7 \]
03

Combine Results into Scientific Notation

Now combine the results from Step 1 and Step 2 to form a new number in scientific notation. You have:\[ 72.0 \times 10^{-7} \]
04

Adjust to Proper Scientific Notation

Since 72.0 is not in the standard form for scientific notation (which is a number between 1 and 10), adjust by moving the decimal one place to the left to get 7.2, and increase the exponent by 1 to compensate:\[ 7.2 \times 10^{-6} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplying Coefficients
In scientific notation, the multiplication process begins with the coefficients. Coefficients are the numbers in front of the powers of ten, such as 9.0 and 8.0 in our problem. The first step in the exercise is to multiply these two coefficients together.
This is like handling simple multiplication with the numbers themselves. For example, when multiplying 9.0 by 8.0, we get 72.0. This part of the calculation does not involve any exponents and functions like normal multiplication.
Multiplying coefficients:
  • Identify the coefficients from each scientific notation term.
  • Multiply these coefficients together as you would in a simple multiplication problem.
  • In our example, the multiplication of the coefficients is: \(9.0 \times 8.0 = 72.0\).
Adding Exponents
Next, we focus on the exponents in the scientific notation. While the multiplication of coefficients dealt with straightforward arithmetic, handling the exponents involves understanding the laws of exponents. For two bases that are the same, you can add their exponents.
In our example:
  • The exponents from the notation \( 10^{-1} \) and \( 10^{-6} \) need to be combined.
  • Since both numbers have a base of 10, allow us to add their exponents directly: \( -1 + (-6) = -7 \).
  • This gives us the combined exponent term of \( 10^{-7} \).
This step is essential as it determines the power of ten that the result will be multiplied by in the scientific notation form. Simplifying and adding the exponents correctly is key to maintaining the right numeric relationship in the resulting scientific format.
Adjusting Scientific Notation
After getting the initial result from multiplying coefficients and adding exponents, the final step is to ensure the number is in the proper scientific notation format.
Scientific notation requires a coefficient that is between 1 and 10. Thus, if the initial multiplication gives a coefficient like 72.0, it will need adjustment.
Here鈥檚 how it鈥檚 done:
  • Identify the coefficient from your previous multiplication (e.g., 72.0 in this example).
  • If this coefficient is not between 1 and 10, adjust by moving the decimal point. For 72.0, move the decimal one place left to get 7.2.
  • To balance the change, increase the exponent by one: from \( 10^{-7} \) to \( 10^{-6} \).
  • So, \( 72.0 \times 10^{-7} \) becomes \( 7.2 \times 10^{-6} \), which is now correctly formatted.
This method ensures that the final expression is both mathematically correct and in the standard scientific notation form, ready for easy interpretation and comparison.

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