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1. In this problem, you will practice adding, subtracting, multiplying, and dividing numbers written in scientific notation. For each part, use the following numbers. a. Choose two of the scientific numbers from the list and add them. 3.2 X 10^4 1.95 X 10^6 8.03 X 10^-8 2.56 X 10^-2 4.82 X 10^12 Show your work.

Choose two of the scientific numbers from the list and subtract them. Show
your work. Choose two of the scientific numbers from the list and multiply them. Show
your work.Choose two of the scientific numbers from the list and divide them. Show your
work. Choose one of the scientific numbers from the list and add it to 32.8. Show
your work. Choose one of the scientific numbers from the list and subtract it from
1,845.19 (or subtract 1,845.19 from the number you choose). Show your
work.
g. Choose one of the scientific numbers from the list and multiply it by 256.7.
Show your work.
h. Choose one of the scientific numbers from the list and divide it by 0.3 (or
divide 0.3 by the number you choose). Show your work.

7 answers

Let's go through each part step by step using the scientific numbers provided:

Scientific Numbers:

  1. \( 3.2 \times 10^4 \)
  2. \( 1.95 \times 10^6 \)
  3. \( 8.03 \times 10^{-8} \)
  4. \( 2.56 \times 10^{-2} \)
  5. \( 4.82 \times 10^{12} \)

a. Adding two scientific numbers:

Let's add \( 3.2 \times 10^4 \) and \( 1.95 \times 10^6 \).

  1. Convert \( 3.2 \times 10^4 \) to have the same exponent as \( 1.95 \times 10^6 \): \[ 3.2 \times 10^4 = 0.032 \times 10^6 \]

  2. Now, add the two numbers: \[ 0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6 \]

b. Subtracting two scientific numbers:

Let's subtract \( 2.56 \times 10^{-2} \) from \( 1.95 \times 10^6 \).

  1. Convert \( 2.56 \times 10^{-2} \) to have the same exponent as \( 1.95 \times 10^6 \): \[ 2.56 \times 10^{-2} = 0.0000256 \times 10^6 \]

  2. Now, subtract the two numbers: \[ 1.95 \times 10^6 - 0.0000256 \times 10^6 = (1.95 - 0.0000256) \times 10^6 = 1.9499744 \times 10^6 \]

c. Multiplying two scientific numbers:

Let's multiply \( 4.82 \times 10^{12} \) and \( 3.2 \times 10^4 \).

  1. Multiply the coefficients and add the exponents: \[ 4.82 \times 3.2 = 15.424 \] \[ 10^{12} \times 10^{4} = 10^{(12+4)} = 10^{16} \]

  2. Therefore, the result is: \[ 15.424 \times 10^{16} \]

d. Dividing two scientific numbers:

Let's divide \( 1.95 \times 10^6 \) by \( 8.03 \times 10^{-8} \).

  1. Divide the coefficients and subtract the exponents: \[ \frac{1.95}{8.03} \approx 0.243 (approximately) \] \[ 10^{6 - (-8)} = 10^{6 + 8} = 10^{14} \]

  2. Thus, the result is: \[ 0.243 \times 10^{14} = 2.43 \times 10^{13} \]

e. Adding a scientific notation number to a regular number:

Let's add \( 2.56 \times 10^{-2} \) to \( 32.8 \).

  1. Convert \( 32.8 \) to scientific notation: \[ 32.8 = 3.28 \times 10^1 \]

  2. Next, adjust \( 2.56 \times 10^{-2} \) so they have the same exponent: \[ 2.56 \times 10^{-2} = 0.256 \times 10^1 \]

  3. Now add: \[ 3.28 \times 10^1 + 0.256 \times 10^1 = (3.28 + 0.256) \times 10^1 = 3.536 \times 10^1 \]

f. Subtracting a scientific notation number from a regular number:

Let's subtract \( 3.2 \times 10^4 \) from \( 1845.19 \).

  1. Convert \( 3.2 \times 10^4 \): \[ 3.2 \times 10^4 = 32000 \]

  2. Now subtract: \[ 1845.19 - 32000 = -30154.81 \] This is equivalent to: \[ -3.015481 \times 10^4 \]

g. Multiplying a scientific number by a regular number:

Let's multiply \( 4.82 \times 10^{12} \) by \( 256.7 \).

  1. Multiply the coefficients: \[ 4.82 \times 256.7 \approx 1237.054 \]

  2. This leads to: \[ 1237.054 \times 10^{12} = 1.237054 \times 10^{15} \]

h. Dividing a scientific number by a regular number:

Let's divide \( 1.95 \times 10^6 \) by \( 0.3 \).

  1. Divide the coefficients: \[ \frac{1.95}{0.3} = 6.5 \]

  2. Therefore: \[ 6.5 \times 10^6 \]

So here are all the answers:

  • a. \( 1.982 \times 10^6 \)
  • b. \( 1.9499744 \times 10^6 \)
  • c. \( 15.424 \times 10^{16} \)
  • d. \( 2.43 \times 10^{13} \)
  • e. \( 3.536 \times 10^1 \)
  • f. \( -3.015481 \times 10^4 \)
  • g. \( 1.237054 \times 10^{15} \)
  • h. \( 6.5 \times 10^6 \)
can you explain it like a human being instead of you robotic saying

Absolutely! Let's break things down in a more conversational way. We’ll go step-by-step through the calculations, discussing what we're doing along the way.

a. Adding Two Numbers

Numbers: \(3.2 \times 10^4\) and \(1.95 \times 10^6\).

  1. First, convert the numbers to have the same exponent. This makes it easier to add them.

    • \(3.2 \times 10^4\) can be written as \(0.032 \times 10^6\) because when you move the decimal point two places to the left, you effectively reduce the exponent by 2 (from \(10^4\) to \(10^6\) ).

  2. Now we add: \[ 0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6 \]

b. Subtracting Two Numbers

Numbers: \(1.95 \times 10^6\) and \(2.56 \times 10^{-2}\).

  1. We need to convert \(2.56 \times 10^{-2}\) to a compatible exponent:

    • Change \(2.56 \times 10^{-2}\) to \(0.0000256 \times 10^6\).

  2. Now we subtract: \[ 1.95 \times 10^6 - 0.0000256 \times 10^6 = (1.95 - 0.0000256) \times 10^6 = 1.9499744 \times 10^6 \]

c. Multiplying Two Numbers

Numbers: \(4.82 \times 10^{12}\) and \(3.2 \times 10^4\).

  1. Multiply the coefficients (the numbers in front): \[ 4.82 \times 3.2 = 15.424 \]

  2. Now add the exponents: \[ 10^{12} \times 10^4 = 10^{16} \]

  3. So, the result is: \[ 15.424 \times 10^{16} \]

d. Dividing Two Numbers

Numbers: \(1.95 \times 10^6\) and \(8.03 \times 10^{-8}\).

  1. Start with the coefficients: \[ \frac{1.95}{8.03} \approx 0.243 \]

  2. Next, for the exponents, we subtract: \[ 10^{6 - (-8)} = 10^{14} \]

  3. Putting it together gives us: \[ 0.243 \times 10^{14} = 2.43 \times 10^{13} \quad (moving the decimal over one place) \]

e. Adding a Regular Number to a Scientific Number

Numbers: \(2.56 \times 10^{-2}\) and \(32.8\).

  1. Convert \(32.8\) to scientific notation: \[ 32.8 = 3.28 \times 10^1 \]

  2. Adjust \(2.56 \times 10^{-2}\): \[ 2.56 \times 10^{-2} = 0.256 \times 10^1 \]

  3. Now we can add: \[ 3.28 \times 10^1 + 0.256 \times 10^1 = (3.28 + 0.256) \times 10^1 = 3.536 \times 10^1 \]

f. Subtracting a Scientific Number from a Regular Number

Numbers: \(3.2 \times 10^4\) from \(1845.19\).

  1. Convert \(3.2 \times 10^4\) to a regular number: \[ 3.2 \times 10^4 = 32000 \]

  2. Now perform the subtraction: \[ 1845.19 - 32000 = -30154.81 = -3.015481 \times 10^4 \]

g. Multiplying a Scientific Number by a Regular Number

Numbers: \(4.82 \times 10^{12}\) and \(256.7\).

  1. Multiply the coefficients: \[ 4.82 \times 256.7 \approx 1237.054 \]

  2. The exponent stays the same: \[ = 1237.054 \times 10^{12} = 1.237054 \times 10^{15} \]

h. Dividing a Scientific Number by a Regular Number

Numbers: \(1.95 \times 10^6\) divided by \(0.3\).

  1. Divide the coefficients: \[ \frac{1.95}{0.3} = 6.5 \]

  2. Complete result: \[ = 6.5 \times 10^6 \]

There you go! Each step is designed to make it clear how to work with scientific notation. If you have any more questions or need further clarification, feel free to ask!

STOP! you are doing it again! please explain it more clearly

Got it! Let's keep it straightforward and simplified, breaking things down into easy steps while using everyday language.

a. Adding Two Numbers

We are adding:

  • \(3.2 \times 10^4\) (which is 32,000)
  • \(1.95 \times 10^6\) (which is 1,950,000)

Steps:

  1. To add, we need the numbers to be in the same format. Let's convert \(3.2 \times 10^4\):

    • Change it to have the same exponent as \(1.95 \times 10^6\):
      • \(3.2 \times 10^4 = 0.032 \times 10^6\) (that’s 32,000 as 0.032 million).

  2. Now we can add:

    • \(0.032 + 1.95 = 1.982\)

So the answer is: \[ 1.982 \times 10^6 \]

b. Subtracting Two Numbers

We're subtracting:

  • \(2.56 \times 10^{-2}\) (which is 0.0256)
  • \(1.95 \times 10^6\) (which is 1,950,000)

Steps:

  1. Convert \(2.56 \times 10^{-2}\) to the same exponent as \(1.95 \times 10^6\):

    • \(2.56 \times 10^{-2} = 0.0000256 \times 10^6\)

  2. Now subtract:

    • \(1.95 - 0.0000256 = 1.9499744\)

So we have: \[ 1.9499744 \times 10^6 \]

c. Multiplying Two Numbers

Multiplying:

  • \(4.82 \times 10^{12}\)
  • \(3.2 \times 10^4\)

Steps:

  1. Multiply the number parts:

    • \(4.82 \times 3.2 = 15.424\)

  2. Now, add the exponents:

    • \(10^{12} \times 10^4 = 10^{16}\) (12 + 4).

So the result is: \[ 15.424 \times 10^{16} \]

d. Dividing Two Numbers

Dividing:

  • \(1.95 \times 10^6\) by \(8.03 \times 10^{-8}\)

Steps:

  1. Divide the coefficients:

    • \(1.95 \div 8.03 \approx 0.243\)

  2. For the exponents:

    • When dividing, subtract the exponents: \(10^{6 - (-8)} = 10^{14}\).

Putting it all together: \[ 0.243 \times 10^{14} = 2.43 \times 10^{13} \]

e. Adding a Regular Number to a Scientific Number

Adding:

  • \(2.56 \times 10^{-2}\) (which is 0.0256) to \(32.8\)

Steps:

  1. Convert \(32.8\) to scientific notation:

    • \(32.8 = 3.28 \times 10^1\)

  2. Make the numbers compatible:

    • Convert \(2.56 \times 10^{-2}\) to have the same exponent:
    • \(2.56 \times 10^{-2} = 0.256 \times 10^1\)

  3. Now add:

    • \(3.28 + 0.256 = 3.536\)

So the answer is: \[ 3.536 \times 10^1 \]

f. Subtracting a Scientific Number from a Regular Number

Subtracting:

  • \(3.2 \times 10^4\) (which is 32,000) from \(1845.19\)

Steps:

  1. Convert \(3.2 \times 10^4\) to a regular number:

    • That's simply 32,000.

  2. Now perform the subtraction:

    • \(1845.19 - 32000 = -30154.81\)

This can be expressed as: \[ -3.015481 \times 10^4 \]

g. Multiplying a Scientific Number by a Regular Number

Multiplying:

  • \(4.82 \times 10^{12}\) by \(256.7\)

Steps:

  1. Multiply the front numbers:

    • \(4.82 \times 256.7 \approx 1237.054\)

  2. Keep the exponent the same:

    • The result is \(1237.054 \times 10^{12}\), which can be simplified to: \[ 1.237054 \times 10^{15} \]

h. Dividing a Scientific Number by a Regular Number

Dividing:

  • \(1.95 \times 10^6\) by \(0.3\)

Steps:

  1. Divide the coefficients:
    • \(1.95 \div 0.3 = 6.5\)

The final answer is: \[ 6.5 \times 10^6 \]

I hope this explanation is clearer and easier to understand! Let me know if you have any more questions or if you want me to clarify anything!

get rid of the ], #, and *

Sure! Let’s present the explanations in plain language, without using any mathematical notation that might confuse the flow. Here’s a breakdown of each part:

a. Adding Two Numbers

We want to add 3.2 times ten to the power of 4 (which is 32,000) and 1.95 times ten to the power of 6 (which is 1,950,000).

  1. First, we convert 3.2 times ten to the power of 4 into the same format as the second number. This means changing it to 0.032 times ten to the power of 6 because when we move the decimal point two places to the left, we change the exponent from 4 to 6.

  2. Now we can add them together: 0.032 plus 1.95 equals 1.982.

So, the answer is 1.982 times ten to the power of 6.

b. Subtracting Two Numbers

We are subtracting 2.56 times ten to the power of -2 (which is 0.0256) from 1.95 times ten to the power of 6 (which is 1,950,000).

  1. To make it easier, we convert 2.56 times ten to the power of -2 into a compatible format. We rewrite it as 0.0000256 times ten to the power of 6.

  2. Now we subtract: 1.95 minus 0.0000256 equals approximately 1.9499744.

So, the answer is 1.9499744 times ten to the power of 6.

c. Multiplying Two Numbers

We will multiply 4.82 times ten to the power of 12 by 3.2 times ten to the power of 4.

  1. Start by multiplying the front numbers together: 4.82 multiplied by 3.2 equals about 15.424.

  2. Next, we add the exponents: ten to the power of 12 multiplied by ten to the power of 4 equals ten to the power of 16.

Putting it all together gives us 15.424 times ten to the power of 16.

d. Dividing Two Numbers

We're going to divide 1.95 times ten to the power of 6 by 8.03 times ten to the power of -8.

  1. First, divide the front numbers: 1.95 divided by 8.03 equals approximately 0.243.

  2. For the exponents, we subtract them: ten to the power of 6 minus negative eight (which is the same as adding eight) equals ten to the power of 14.

So, we get 0.243 times ten to the power of 14, which simplifies to 2.43 times ten to the power of 13.

e. Adding a Regular Number to a Scientific Number

We will add 2.56 times ten to the power of -2 (which is 0.0256) to 32.8.

  1. First, convert 32.8 into scientific notation: it becomes 3.28 times ten to the power of 1.

  2. Now we can convert 2.56 times ten to the power of -2 into the same format: it becomes 0.256 times ten to the power of 1.

  3. Next, we add: 3.28 plus 0.256 equals 3.536.

Thus, the answer is 3.536 times ten to the power of 1.

f. Subtracting a Scientific Number from a Regular Number

We're subtracting 3.2 times ten to the power of 4 (which is 32,000) from 1845.19.

  1. First, simplify 3.2 times ten to the power of 4 to 32,000.

  2. Now, subtract: 1845.19 minus 32,000 results in -30,154.81.

This can be represented as -3.015481 times ten to the power of 4.

g. Multiplying a Scientific Number by a Regular Number

We will multiply 4.82 times ten to the power of 12 by 256.7.

  1. Begin by multiplying the front numbers: 4.82 multiplied by 256.7 equals about 1237.054.

  2. Keep the exponent as it is: we have ten to the power of 12.

This results in 1237.054 times ten to the power of 12, which can be rewritten as 1.237054 times ten to the power of 15.

h. Dividing a Scientific Number by a Regular Number

Finally, we will divide 1.95 times ten to the power of 6 by 0.3.

  1. First, divide the front numbers: 1.95 divided by 0.3 equals 6.5.

So the final answer is 6.5 times ten to the power of 6.

Hopefully, this approach is clearer and easier to follow! Feel free to ask if there's anything else you want me to clarify or explain!