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Multiply the expression on the left and match them with their appropriate result.%0D%0A%0D%0AShow your work.%0D%0A%0D%0A(3 points)%0D%0APut responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.%0D%0A(2 × 10−3) ⋅ (3 × 107)%0D%0A(%0D%0A2%0D%0A %0D%0A×%0D%0A %0D%0A10%0D%0A−%0D%0A3%0D%0A)%0D%0A %0D%0A⋅%0D%0A %0D%0A(%0D%0A3%0D%0A %0D%0A×%0D%0A %0D%0A10%0D%0A7%0D%0A)%0D%0A(9.1 × 109) ⋅ (2.6 × 102)%0D%0A(%0D%0A9.1%0D%0A %0D%0A×%0D%0A %0D%0A10%0D%0A9%0D%0A)%0D%0A %0D%0A⋅%0D%0A %0D%0A(%0D%0A2.6%0D%0A %0D%0A×%0D%0A %0D%0A10%0D%0A2%0D%0A)%0D%0A(2.366 × 1012)%0D%0A(%0D%0A2.366%0D%0A %0D%0A×%0D%0A %0D%0A10%0D%0A12%0D%0A)%0D%0A(6.77 × 10−12) ⋅ (1.12 × 10−6)%0D%0A(%0D%0A6.77%0D%0A %0D%0A×%0D%0A %0D%0A10%0D%0A−%0D%0A12%0D%0A)%0D%0A %0D%0A⋅%0D%0A %0D%0A(%0D%0A1.12%0D%0A %0D%0A×%0D%0A %0D%0A10%0D%0A−%0D%0A6%0D%0A)

9 answers

To multiply the given expressions in scientific notation, we follow the rule that states:

\[ (a \times 10^m) \cdot (b \times 10^n) = (a \cdot b) \times 10^{m+n} \]

Let's solve each expression step by step.

First Expression:

\((2 \times 10^{-3}) \cdot (3 \times 10^{7})\)

Step 1: Multiply the coefficients:

\[ 2 \times 3 = 6 \]

Step 2: Add the exponents:

\[ -3 + 7 = 4 \]

Result:

\[ 6 \times 10^{4} \]

Second Expression:

\((9.1 \times 10^{9}) \cdot (2.6 \times 10^{2})\)

Step 1: Multiply the coefficients:

\[ 9.1 \times 2.6 = 23.66 \]

Step 2: Add the exponents:

\[ 9 + 2 = 11 \]

Result:

\[ 23.66 \times 10^{11} \]

Third Expression:

\((2.366 \times 10^{12})\)

This expression is already simplified. The result is:

\[ 2.366 \times 10^{12} \]

Fourth Expression:

\((6.77 \times 10^{-12}) \cdot (1.12 \times 10^{-6})\)

Step 1: Multiply the coefficients:

\[ 6.77 \times 1.12 = 7.5774 \]

Step 2: Add the exponents:

\[ -12 + (-6) = -18 \]

Result:

\[ 7.5774 \times 10^{-18} \]

Summary of Results:

  1. \( (2 \times 10^{-3}) \cdot (3 \times 10^{7}) = 6 \times 10^{4} \)
  2. \( (9.1 \times 10^{9}) \cdot (2.6 \times 10^{2}) = 23.66 \times 10^{11} \)
  3. \( 2.366 \times 10^{12} \) (already simplified)
  4. \( (6.77 \times 10^{-12}) \cdot (1.12 \times 10^{-6}) = 7.5774 \times 10^{-18} \)

Now match each expression with its appropriate result as needed.

no u match

Sure! Let's match each of the results with their corresponding expressions:

  1. Expression: \( (2 \times 10^{-3}) \cdot (3 \times 10^{7}) \)
    Result: \( 6 \times 10^{4} \)

  2. Expression: \( (9.1 \times 10^{9}) \cdot (2.6 \times 10^{2}) \)
    Result: \( 23.66 \times 10^{11} \)

  3. Expression: \( (2.366 \times 10^{12}) \)
    Result: \( 2.366 \times 10^{12} \) (itself)

  4. Expression: \( (6.77 \times 10^{-12}) \cdot (1.12 \times 10^{-6}) \)
    Result: \( 7.5774 \times 10^{-18} \)

So the matched pairs are:

  • \((2 \times 10^{-3}) \cdot (3 \times 10^{7})\) matches with \(6 \times 10^{4}\)
  • \((9.1 \times 10^{9}) \cdot (2.6 \times 10^{2})\) matches with \(23.66 \times 10^{11}\)
  • \((2.366 \times 10^{12})\) matches with \(2.366 \times 10^{12}\)
  • \((6.77 \times 10^{-12}) \cdot (1.12 \times 10^{-6})\) matches with \(7.5774 \times 10^{-18}\)

If you need further assistance with how to input them or any additional information, please let me know!

show all the steps but dumb them down

Sure! Let’s break down each multiplication step by step in a simple way.

Expression 1: \((2 \times 10^{-3}) \cdot (3 \times 10^{7})\)

Step 1: Multiply the numbers in front (coefficients):

  • \(2 \times 3 = 6\)

Step 2: Add the powers of ten (exponents):

  • \(-3 + 7 = 4\)

Step 3: Combine the results:

  • Final result: \(6 \times 10^{4}\)

Expression 2: \((9.1 \times 10^{9}) \cdot (2.6 \times 10^{2})\)

Step 1: Multiply the numbers in front (coefficients):

  • \(9.1 \times 2.6 = 23.66\)
    (You can use a calculator for this step!)

Step 2: Add the powers of ten (exponents):

  • \(9 + 2 = 11\)

Step 3: Combine the results:

  • Final result: \(23.66 \times 10^{11}\)

Expression 3: \((2.366 \times 10^{12})\)

This one is already simplified!
So, the result is simply:

  • Final result: \(2.366 \times 10^{12}\)

Expression 4: \((6.77 \times 10^{-12}) \cdot (1.12 \times 10^{-6})\)

Step 1: Multiply the numbers in front (coefficients):

  • \(6.77 \times 1.12 = 7.5774\)
    (Again, you can use a calculator for this step!)

Step 2: Add the powers of ten (exponents):

  • \(-12 + (-6) = -18\)
    (When you add negative numbers, it's like subtracting, so you get -18.)

Step 3: Combine the results:

  • Final result: \(7.5774 \times 10^{-18}\)

Summary of Results

  1. First expression: \((2 \times 10^{-3}) \cdot (3 \times 10^{7}) = 6 \times 10^{4}\)
  2. Second expression: \((9.1 \times 10^{9}) \cdot (2.6 \times 10^{2}) = 23.66 \times 10^{11}\)
  3. Third expression: \((2.366 \times 10^{12}) = 2.366 \times 10^{12}\)
  4. Fourth expression: \((6.77 \times 10^{-12}) \cdot (1.12 \times 10^{-6}) = 7.5774 \times 10^{-18}\)

This is the simplified process for multiplying the scientific notation expressions. If you have any further questions on any step, feel free to ask!

no words

Got it! Here are the steps without words:

Expression 1: \((2 \times 10^{-3}) \cdot (3 \times 10^{7})\)

  1. \(2 \times 3 = 6\)
  2. \(-3 + 7 = 4\)
  3. Result: \(6 \times 10^{4}\)

Expression 2: \((9.1 \times 10^{9}) \cdot (2.6 \times 10^{2})\)

  1. \(9.1 \times 2.6 = 23.66\)
  2. \(9 + 2 = 11\)
  3. Result: \(23.66 \times 10^{11}\)

Expression 3: \((2.366 \times 10^{12})\)

  1. Result: \(2.366 \times 10^{12}\)

Expression 4: \((6.77 \times 10^{-12}) \cdot (1.12 \times 10^{-6})\)

  1. \(6.77 \times 1.12 = 7.5774\)
  2. \(-12 + (-6) = -18\)
  3. Result: \(7.5774 \times 10^{-18}\)

Final Results:

  1. \(6 \times 10^{4}\)
  2. \(23.66 \times 10^{11}\)
  3. \(2.366 \times 10^{12}\)
  4. \(7.5774 \times 10^{-18}\)
expresion 2 is wroug

Apologies for the mistake in the multiplication for Expression 2. Let's redo it step by step correctly.

Expression 2: \((9.1 \times 10^{9}) \cdot (2.6 \times 10^{2})\)

Step 1: Multiply the numbers in front (coefficients):

\[ 9.1 \times 2.6 = 23.66 \]

(I made a mistake here previously; let's calculate it without skipping!)

Calculating it correctly:

\[ 9.1 \cdot 2.6 = 23.66 \]

Step 2: Add the powers of ten (exponents):

\[ 9 + 2 = 11 \]

Step 3: Combine the results:

Correct Result:
Since we have \(23.66\), it must be expressed in proper scientific notation, which would be:

\[ 2.366 \times 10^{12} \]

Correct Summary of Results:

  1. First expression: \(6 \times 10^{4}\)
  2. Second expression: \(2.366 \times 10^{12}\) (corrected)
  3. Third expression: \(2.366 \times 10^{12}\)
  4. Fourth expression: \(7.5774 \times 10^{-18}\)

Thank you for your patience! If you need further clarifications or calculations, let me know!