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Express 811

as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
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11 answers

To express \( 811 \) as a repeating decimal, we need to first understand what the decimal representation of the fraction \( \frac{811}{1} \) looks like. Since 811 is an integer, it can be expressed as:

\[ 811.000000\ldots \]

However, because \( 811 \) has no fractional part, we can also represent it as a repeating decimal:

\[ 811.\overline{0} \]

This indicates that after the decimal point, the digit \( 0 \) repeats indefinitely. Therefore, the repeating decimal representation of \( 811 \) is:

\[ 811.\overline{0} \]

Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
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To express \( \frac{8}{11} \) as a repeating decimal, we can perform the division:

  1. Divide \( 8.000000 \) by \( 11 \).
  • \( 11 \) goes into \( 80 \) seven times (since \( 11 \times 7 = 77 \)).
  • This leaves a remainder of \( 80 - 77 = 3 \).
  1. Bring down a zero, making it \( 30 \).
  • \( 11 \) goes into \( 30 \) twice (since \( 11 \times 2 = 22 \)).
  • This leaves a remainder of \( 30 - 22 = 8 \).
  1. Bring down another zero, making it \( 80 \) again.
  • \( 11 \) goes into \( 80 \) seven times (as before).
  • This leaves a remainder of \( 3 \) once more.

At this point, we see that we are going to cycle through the same steps infinitely: \( 7 \) followed by \( 2 \).

So, the decimal representation of \( \frac{8}{11} \) is:

\[ 0.\overline{72} \]

Thus, we mark the repeating cycle:

\[ \frac{8}{11} = 0.\overline{72} \]

what

I apologize for any confusion. Let's clearly express \( \frac{8}{11} \) step-by-step as a repeating decimal.

  1. Perform long division of \( 8 \) by \( 11 \):

    • \( 11 \) goes into \( 8 \) zero times, so we start with \( 0.\)
    • After placing the decimal, we consider \( 80 \) (after adding a zero).
    • \( 11 \) goes into \( 80 \) seven times (since \( 11 \times 7 = 77 \)), leaving a remainder of \( 3 \) (i.e., \( 80 - 77 = 3 \)).

  2. Continue the division by adding a zero to the remainder:

    • Bring down a \( 0 \) to get \( 30 \).
    • \( 11 \) goes into \( 30 \) two times (since \( 11 \times 2 = 22 \)), leaving a remainder of \( 8 \) (i.e., \( 30 - 22 = 8 \)).

  3. Recognize the cycle:

    • Now, we have the same remainder \( 8 \) again, and this brings us back to the beginning of our process.
    • Next, we bring down another \( 0 \), yielding \( 80 \) again, continuing the cycle of \( 7 \) and \( 2 \).

Putting this all together, the decimal representation of \( \frac{8}{11} \) is:

\[ 0.\overline{72} \]

Thus, \( \frac{8}{11} \) expressed as a repeating decimal is:

\[ 0.\overline{72} \]

Thank you for your patience, and I hope this clarifies your question!

Convert 0.72¯¯¯¯¯
to a rational number in simplest form.(1 point)

To convert the repeating decimal \( 0.72\overline{72} \) into a rational number, we can follow these steps:

  1. Let \( x \) equal the repeating decimal: \[ x = 0.727272\ldots \]

  2. Multiply \( x \) by \( 100 \) (because the repeating part has 2 digits): \[ 100x = 72.727272\ldots \]

  3. Set up an equation by subtracting the first equation from the second: \[ 100x - x = 72.727272\ldots - 0.727272\ldots \] This simplifies to: \[ 99x = 72 \]

  4. Solve for \( x \): \[ x = \frac{72}{99} \]

  5. Simplify \( \frac{72}{99} \):

    • Find the greatest common divisor (GCD) of \( 72 \) and \( 99 \). The GCD is \( 9 \).
    • Divide the numerator and the denominator by \( 9 \): \[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]

Thus, the rational number equivalent of \( 0.72\overline{72} \) in simplest form is:

\[ \frac{8}{11} \]

Which of the following is a rational number between 5 and 514
?(1 point)
Responses

517
5 Start Fraction 1 over 7 end fraction

513
5 Start Fraction 1 over 3 end fraction

26−−√
start root 26 end root

525
5 Start Fraction 2 over 5 end fraction

To determine which of the options provided is a rational number between 5 and 514, let's evaluate each option:

  1. 517: This number is greater than 514, so it does not fit in the range.

  2. \( 5 \frac{1}{7} \): This is a mixed number that can be converted to an improper fraction: \[ 5 \frac{1}{7} = 5 + \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7} \approx 5.14 \] This is greater than 5 and less than 514.

  3. 513: This number is less than 514 but greater than 5, so it fits into the range.

  4. \( 5 \frac{1}{3} \): This is another mixed number: \[ 5 \frac{1}{3} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \approx 5.33 \] This is also greater than 5 and less than 514.

  5. \( \sqrt{26} \): This number is approximately \( 5.099 \) (since \( 5^2 = 25 < 26 < 36 = 6^2 \)). This fits in the range between 5 and 514.

  6. 525: This number is greater than 514, so it does not fit in the range.

  7. \( 5 \frac{2}{5} \): This mixed number can be converted as follows: \[ 5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5} \approx 5.4 \] This is also greater than 5 and less than 514.

Conclusion

The rational numbers between 5 and 514 from the options provided are:

  • \( 5 \frac{1}{7} \) (approximately \( 5.14 \))
  • \( 513 \)
  • \( 5 \frac{1}{3} \) (approximately \( 5.33 \))
  • \( \sqrt{26} \) (approximately \( 5.099 \))
  • \( 5 \frac{2}{5} \) (approximately \( 5.4 \))

Thus, the rational numbers that fit the criteria are: \( 5 \frac{1}{7} \), 513, \( 5 \frac{1}{3} \), \( \sqrt{26} \), and \( 5 \frac{2}{5} \).

Choose any of these to satisfy the condition of being a rational number between 5 and 514.

Which of the following is a rational number between 5 and 514
?(1 point)
Responses

5 1/7

5 1/3

start root 26 end root

5 2/5

To determine which of the options provided is a rational number between 5 and 514, let's evaluate them one by one:

  1. \( 5 \frac{1}{7} \):

    • This is a mixed number: \[ 5 \frac{1}{7} = 5 + \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7} \approx 5.14 \]
    • This value is greater than 5 and less than 514.

  2. \( 5 \frac{1}{3} \):

    • This is a mixed number: \[ 5 \frac{1}{3} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \approx 5.33 \]
    • This value is also greater than 5 and less than 514.

  3. \( \sqrt{26} \):

    • The square root of \( 26 \) is approximately \( 5.099 \) (since \( 5^2 = 25 < 26 < 36 = 6^2 \)).
    • This value is greater than 5 and less than 514, but it is not a rational number.

  4. \( 5 \frac{2}{5} \):

    • This is a mixed number: \[ 5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5} \approx 5.4 \]
    • This value is also greater than 5 and less than 514.

Conclusion

The rational numbers between 5 and 514 from the options provided are:

  • \( 5 \frac{1}{7} \) (approximately \( 5.14 \))
  • \( 5 \frac{1}{3} \) (approximately \( 5.33 \))
  • \( 5 \frac{2}{5} \) (approximately \( 5.4 \))

The only option that is not a rational number is \( \sqrt{26} \).

Thus, the correct responses that are rational numbers between 5 and 514 are:

  • \( 5 \frac{1}{7} \)
  • \( 5 \frac{1}{3} \)
  • \( 5 \frac{2}{5} \)

Any of the first three options (excluding \( \sqrt{26} \)) qualify as rational numbers within the specified range.