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When multiplying two numbers in scientific notation, what operations do you use on the coefficients and the exponents of the powers of 10?

1. Add the coefficients and multiply the exponents of the powers of 10.
2. Multiply the coefficients and add the exponents of the powers of 10.
3. Add the coefficients and add the exponents of the powers of 10.
4. Multiply the coefficients and multiply the exponents of the powers of 10.

7 answers

When multiplying two numbers in scientific notation, you use the following operations:

  1. Multiply the coefficients and add the exponents of the powers of 10.

For example, if you have \( (a \times 10^m) \) and \( (b \times 10^n) \), then the product will be \( (a \times b) \times 10^{(m+n)} \).

A tape dispenser holds 1.14 times 10^4mm of tape that is 12.7mm wide. Multiply the two numbers to find the total area in square millimeters of tape. Write the product in scientific notation.

To find the total area in square millimeters of the tape, you can multiply the length of the tape by its width.

Given:

  • Length of tape: \( 1.14 \times 10^4 \) mm
  • Width of tape: \( 12.7 \) mm

First, multiply the coefficients:

\[ 1.14 \times 12.7 = 14.478 \]

Next, for the scientific notation, multiply the powers of 10. Since the width is a whole number (which we can express as \( 12.7 = 12.7 \times 10^0 \)), we can express the multiplication of powers of 10 as:

\[ 10^4 \times 10^0 = 10^4 \]

Putting it together:

\[ \text{Area} = 14.478 \times 10^4 \]

Now, we need to convert \( 14.478 \) into scientific notation. \( 14.478 \) can be expressed as \( 1.4478 \times 10^1 \).

So, the area becomes:

\[ 14.478 \times 10^4 = (1.4478 \times 10^1) \times 10^4 \]

Adding the exponents:

\[ = 1.4478 \times 10^{1 + 4} = 1.4478 \times 10^5 \]

Thus, the total area in square millimeters of tape is:

\[ \boxed{1.4478 \times 10^5} \text{ mm}^2 \]

A sugar cube has a volume of about 4.9 ml. What is the total volume in milliliters of 1.5 times 10^6 sugar cubes?

To find the total volume of \(1.5 \times 10^6\) sugar cubes, you can multiply the volume of one sugar cube by the number of sugar cubes.

Given:

  • Volume of one sugar cube = \(4.9\) ml
  • Number of sugar cubes = \(1.5 \times 10^6\)

First, multiply the coefficients: \[ 1.5 \times 4.9 = 7.35 \]

Next, for the scientific notation, since the second number \(1.5 \times 10^6\) does not involve a power with a base in the volume component \(4.9\), we can write: \[ 10^0 \text{ for the volume} \]

So, multiplying the powers of 10: \[ 10^6 \text{ (from the sugar cubes)} \]

Putting it all together, the total volume becomes: \[ 7.35 \times 10^0 \times 10^6 = 7.35 \times 10^{6} \]

Thus, the total volume in milliliters of \(1.5 \times 10^6\) sugar cubes is: \[ \boxed{7.35 \times 10^6} \text{ ml} \]

Multiply 0.75 by 2.8 times 10^-5.
2.1 times 10^-7
2.1 times 10^-6
2.1 times 10^-4
2.1 times 10^-5

To multiply \(0.75\) by \(2.8 \times 10^{-5}\), you can follow these steps:

  1. Multiply the coefficients:
    \(0.75 \times 2.8 = 2.1\)

  2. Keep the power of 10:
    The power of \(10\) remains as is: \(10^{-5}\)

Putting it together, the product is: \[ 2.1 \times 10^{-5} \]

Thus, the answer is: \[ \boxed{2.1 \times 10^{-5}} \]