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convert 0.7 (theres a bar over 7) to a fraction.

5 answers

To convert the repeating decimal \(0.7\overline{7}\) to a fraction, follow these steps:

  1. Let \(x = 0.7\overline{7}\).

  2. To eliminate the repeating part, multiply \(x\) by 10 (which shifts the decimal point one place to the right): \[ 10x = 7.7\overline{7} \]

  3. Now, subtract the original \(x\) from this equation: \[ 10x - x = 7.7\overline{7} - 0.7\overline{7} \] Simplifying this gives: \[ 9x = 7 \]

  4. Now, solve for \(x\): \[ x = \frac{7}{9} \]

Thus, \(0.7\overline{7}\) as a fraction is \(\frac{7}{9}\).

which of the following is an irrational number between 4 and 8?
1. square root of 12
2. square root of 5
3. square root of 8
4. square root of 17

To determine which of the given options is an irrational number between 4 and 8, we can evaluate each one:

  1. Square root of 12: \[ \sqrt{12} \approx 3.46 \quad \text{(not between 4 and 8, and is irrational)} \]

  2. Square root of 5: \[ \sqrt{5} \approx 2.24 \quad \text{(not between 4 and 8, and is irrational)} \]

  3. Square root of 8: \[ \sqrt{8} = 2\sqrt{2} \approx 2.83 \quad \text{(not between 4 and 8, and is irrational)} \]

  4. Square root of 17: \[ \sqrt{17} \approx 4.12 \quad \text{(between 4 and 8, and is irrational)} \]

Among the options provided, \(\sqrt{17}\) is the only irrational number that falls within the range of 4 and 8.

Which of the following rational numbers is not between the square root of 3 and the square root of 6
1. 1.8
2. the square root of 4
3. 1.5
4. 2

To determine which of the given rational numbers is not between \(\sqrt{3}\) and \(\sqrt{6}\), we first need to find approximate values for \(\sqrt{3}\) and \(\sqrt{6}\):

  • Calculate \(\sqrt{3}\): \[ \sqrt{3} \approx 1.732 \]

  • Calculate \(\sqrt{6}\): \[ \sqrt{6} \approx 2.449 \]

Now we can establish the interval:

  • \(\sqrt{3} \approx 1.732\)
  • \(\sqrt{6} \approx 2.449\)

The rational numbers we need to evaluate are:

  1. 1.8: \[ 1.732 < 1.8 < 2.449 \quad \text{(between)} \]

  2. \(\sqrt{4}\): \[ \sqrt{4} = 2 \quad \text{(between)} \]

  3. 1.5: \[ 1.5 < 1.732 \quad \text{(not between)} \]

  4. 2: \[ 1.732 < 2 < 2.449 \quad \text{(between)} \]

The only number from the list that is not between \(\sqrt{3}\) and \(\sqrt{6}\) is 1.5.