Question

directions:%20Use%20this%20worksheet%20to%20record%20your%20answers%20to%20the%20three%20activities%20that%20make%20up%20the%20Scientific%20Notation%20Portfolio.%20When%20you%20are%20finished,%20save%20this%20worksheet%20with%20your%20answers%20and%20submit%20it%20for%20a%20portfolio%20grade.%0APART%201.%20In%20this%20problem,%20you%20will%20practice%20adding,%20subtracting,%20multiplying,%20and%20dividing%20numbers%20written%20in%20scientific%20notation.%20For%20each%20part,%20use%20the%20following%20numbers.%0A3.2%20%C3%97%2010%E2%82%AC%0A1.95%20%C3%97%2010%E2%82%AC%0A8.03%20%C3%97%2010-9%0A2.56%20%C3%97%2010-*%0AComplete%20the%20Following%20Problems-%20Show%20all%20work%20when%20adding.%20subtracting,%20multiplying%20or%20dividing.

Answer 1

Sure! Let's go through the operations step by step for the numbers in scientific notation that you've provided:

PART 1: Operations with Scientific Notation

1. Adding and Subtracting:

   • To add or subtract numbers in scientific notation, they must have the same exponent (power of 10). If they do not, adjust one or both to have a common exponent.

   • Example Operations:

     • Problem 1: \(3.2 \times 10^a + 1.95 \times 10^a\)

     • Since both have the same exponent, you can directly add the coefficients: \[ (3.2 + 1.95) \times 10^a = 5.15 \times 10^a \]

     • Problem 2: \(8.03 \times 10^{-9} + 2.56 \times 10^{-9}\)

       • Add the coefficients: \[ (8.03 + 2.56) \times 10^{-9} = 10.59 \times 10^{-9} \]

       • The result can be expressed as: \[ 1.059 \times 10^{-8} (\text{by shifting the decimal}) \]

2. Multiplying:

   • When multiplying numbers in scientific notation, multiply the coefficients and add the exponents.

   • Example Operations:

     • Problem 3: \(3.2 \times 10^a \times 1.95 \times 10^a\) \[ = (3.2 \times 1.95) \times 10^{a+a} = 6.24 \times 10^{2a} \]

     • Problem 4: \(8.03 \times 10^{-9} \times 2.56 \times 10^b\) \[ = (8.03 \times 2.56) \times 10^{-9 + b} = 20.5488 \times 10^{-9+b} \]

3. Dividing:

   • When dividing numbers in scientific notation, divide the coefficients and subtract the exponent of the denominator from the exponent of the numerator.

   • Example Operations:

     • Problem 5: \(\frac{3.2 \times 10^a}{1.95 \times 10^a}\) \[ = \left(\frac{3.2}{1.95}\right) \times 10^{a-a} = 1.641 \times 10^0 = 1.641 \]

     • Problem 6: \(\frac{8.03 \times 10^{-9}}{2.56 \times 10^b}\) \[ = \left(\frac{8.03}{2.56}\right) \times 10^{-9 - b} \approx 3.137 \times 10^{-9-b} \]

Summary of Results:

1. Addition:

   • \(5.15 \times 10^a\)
   • \(1.059 \times 10^{-8}\)

2. Multiplication:

   • \(6.24 \times 10^{2a}\)
   • \(20.5488 \times 10^{-9+b}\)

3. Division:

   • \(1.641\)
   • \(3.137 \times 10^{-9-b}\)

Make sure to replace \(a\) and \(b\) with their actual values from your worksheet (if applicable) for exact answers.

Be sure to write down each step in your worksheet, showing how you simplified each part according to the rules of scientific notation.