Problem 71 - Elementary and Intermediate Algebra

Question

Use scientific notation to perform the calculations. Give all answers in scientific notation and standard notation. ( 0.0000000056)\((5,500,000)\)

Step-by-Step Solution

Answer

Scientific notation: 3.08 x 10^{-2}; Standard notation: 0.0308.

Step 1: Identify Numbers in Scientific Notation

First, let's express the given numbers in scientific notation. The number 0.0000000056 can be expressed as 5.6 x 10^{-9}, and the number 5,500,000 can be expressed as 5.5 x 10^6.

Step 2: Perform Multiplication

Now multiply the numbers in scientific notation: (5.6 x 10^{-9}) x (5.5 x 10^6). Multiply the coefficients (5.6 and 5.5) separately from the powers of ten: 5.6 x 5.5 = 30.8, and for the powers of ten: 10^{-9} x 10^6 = 10^{-3}.

Step 3: Combine the Results

Combine the results from the multiplication: 30.8 x 10^{-3}. This needs to be adjusted to stay in scientific notation.

Step 4: Adjust Scientific Notation

Adjust 30.8 to be between 1 and 10, which means expressing it as 3.08 x 10^1. So, we have (3.08 x 10^1) x (10^{-3}).

Step 5: Final Scientific Notation

Multiply the powers of ten: 3.08 x 10^{1} x 10^{-3} = 3.08 x 10^{-2}. This is the final answer in scientific notation.

Step 6: Convert to Standard Notation

Convert the scientific notation 3.08 x 10^{-2} to standard notation: Move the decimal point 2 places to the left, resulting in 0.0308.

Key Concepts

Multiplication of Powers of Ten

Multiplication of powers of ten is a straightforward concept once you get the hang of it. When multiplying numbers in scientific notation, you handle the coefficients and powers of ten separately. This step-by-step approach simplifies complex multiplication. Let's break it down.

• First, multiply the coefficients as you would with any regular numbers. For instance, using the example (5.6 x 10^{-9}) and (5.5 x 10^6), the multiplication of coefficients is simply 5.6 x 5.5, which equals 30.8.


• Next, focus on the powers of ten. The multiplication rule is simple: add the exponents of the base 10s. Thus, 10^{-9} x 10^6 equals 10^{-3} since -9 + 6 = -3.

By treating these two parts separately, you avoid unnecessary complications. Then, combine your results to form a complete answer in scientific notation, like 30.8 x 10^{-3} in our example.

Conversion Between Scientific and Standard Notation

Converting numbers between scientific and standard notation is a critical skill in handling numbers efficiently. It helps manage very large or very small numbers without burying you in zeros.

• Scientific notation is structured as a coefficient between 1 and 10, multiplied by 10 raised to an exponent. For example, the number 5,500,000 becomes 5.5 x 10^6.


• To convert back to standard notation, adjust the decimal position according to the exponent: moving it right for positive exponents and left for negative ones.

In our example, the scientific notation 3.08 x 10^{-2} converts to standard form by moving the decimal two places left, resulting in 0.0308. This method enhances both clarity and simplicity.

Expressing Numbers in Scientific Notation

Expressing numbers in scientific notation provides a streamlined way of handling vast ranges of magnitudes without confusion. Here's how you can convert numbers to this concise form:

• Identify the significant figures by locating the first non-zero digit.


• Position the decimal after this first non-zero digit to create a number between 1 and 10. For instance, with 0.0000000056, the significant number is 5.6.


• Count how many places the decimal was moved to reach its new position; this number is your exponent. Since 5.6 originates from 0.0000000056, the decimal moved 9 places to the right; hence the exponent is -9.

The number is then expressed as 5.6 x 10^{-9}. Understanding this representation empowers you to handle scientific data confidently and efficiently.