Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar
To express 8/11 as a repeating decimal, we need to perform long division:
0.-----
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11 | 8.000000
- 0
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80
- 77
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30
- 22
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80
- 77
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30
- 22
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80
- 77
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30
- 22
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80
- 77
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30
- 22
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80
- 77
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30
- 22
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80
- 77
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30
- 22
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80
- 77
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30
- 22
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80
- 77
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30
- 22
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As we can see, the division keeps repeating the sequence 30 - 22. Hence, 8/11 expressed as a repeating decimal is 0.72727272... or 0.72¯¯¯¯¯¯¯¯.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form
Let's represent 0.72¯¯¯¯¯ as a variable, say x.
To eliminate the decimal part of the repeating decimal, we can multiply x by 100, which will shift the repeating part to the left of the decimal point.
100x = 72.727272...
Now, we subtract x from 100x to eliminate the repeating part:
100x - x = 72.727272... - 0.727272...
99x = 72
Simplifying this equation, we divide both sides by 99:
x = 72/99
The numerator and denominator of 72/99 share a common factor of 9, so we can simplify further:
x = 8/11
Therefore, the rational number equivalent to 0.72¯¯¯¯¯ in simplest form is 8/11.
Which of the following is a rational number between 5 and 5 1/4
To find a rational number between 5 and 5 1/4, we can convert the mixed number 5 1/4 to an improper fraction.
5 1/4 can be written as (4 * 5 + 1) / 4 = 21/4.
Now, we need to find a rational number between 5 and 21/4.
One possible rational number between them is (5 + 21/4) / 2 = (20 + 21) / 4 = 41/4.
Therefore, 41/4 is a rational number between 5 and 5 1/4.
incorrect
Apologies for the mistake. Let's find the correct rational number between 5 and 5 1/4.
To find a rational number between 5 and 5 1/4, we need to determine a common denominator for 5 and 5 1/4. The denominator for 5 is 1, and the denominator for 5 1/4 is 4. To find a common denominator, we multiply the denominator of 5 by the denominator of 5 1/4, resulting in 1 * 4 = 4.
Now, let's convert 5 to have a denominator of 4 by multiplying both the numerator and denominator by 4:
5 * 4/1 * 4 = 20/4
So, 5 can be represented as 20/4.
Now, we have the rational numbers 20/4 and 5 1/4, and we want to find a rational number between them.
Finding the average of these two rational numbers will give us the desired rational number:
(20/4 + 21/4)/2 = 41/8
Therefore, the rational number between 5 and 5 1/4 is 41/8.
Which of the following is a rational number between 5 and 514?(1 point)
Responses
513
5 Start Fraction 1 over 3 end fraction
517
5 Start Fraction 1 over 7 end fraction
26−−√
start root 26 end root
525
5 start fraction 2 over 5 end fraction
To find a rational number between 5 and 514, we need to look for options that fall between these two numbers.
Out of the given options, the only rational number that lies between 5 and 514 is 513.
Therefore, the correct answer is 513.
What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction
To find a rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\), we need to look for a number that can be expressed as \(\frac{a}{7}\), where \(a\) is an integer.
Let's convert \(\sqrt{7}\) to a number with a denominator of 7:
\(\sqrt{7} = \frac{\sqrt{7} \times 7}{7} = \frac{7\sqrt{7}}{7} = \frac{7}{7} \times \frac{\sqrt{7}}{1} = 1 \times \frac{\sqrt{7}}{1} = \frac{\sqrt{7}}{1}\)
Similarly, let's convert \(\sqrt{8}\) to a number with a denominator of 7:
\(\sqrt{8} = \frac{\sqrt{8} \times 7}{7} = \frac{7\sqrt{8}}{7} = \frac{7}{7} \times \frac{\sqrt{8}}{1} = 1 \times \frac{\sqrt{8}}{1} = \frac{\sqrt{8}}{1}\)
Now, we need to find a rational number with a denominator of 7 between \(\frac{\sqrt{7}}{1}\) and \(\frac{\sqrt{8}}{1}\). One possible solution is their average:
\(\frac{\frac{\sqrt{7}}{1} + \frac{\sqrt{8}}{1}}{2} = \frac{\sqrt{7} + \sqrt{8}}{2}\)
However, this answer is not in the form of a fraction with a denominator of 7. To achieve that, we can multiply the numerator and denominator by \(\sqrt{7} - \sqrt{8}\):
\(\frac{\sqrt{7} + \sqrt{8}}{2} \times \frac{\sqrt{7} - \sqrt{8}}{\sqrt{7} - \sqrt{8}} = \frac{{(\sqrt{7} + \sqrt{8})(\sqrt{7} - \sqrt{8})}}{2(\sqrt{7} - \sqrt{8})}\)
Using the difference of squares identity (\(a^2 - b^2 = (a + b)(a - b)\)), we can simplify the numerator:
\(\frac{{7 - 8}}{2(\sqrt{7} - \sqrt{8})} = \frac{{-1}}{2(\sqrt{7} - \sqrt{8})}\)
Therefore, the rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\) can be expressed as \(\frac{-1}{2(\sqrt{7} - \sqrt{8})}\) in improper fraction form.
simplify the answer
To simplify \(\frac{-1}{2(\sqrt{7} - \sqrt{8})}\), we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of \(\sqrt{7} - \sqrt{8}\), which is \(\sqrt{7} + \sqrt{8}\):
\(\frac{-1}{2(\sqrt{7} - \sqrt{8})} \times \frac{\sqrt{7} + \sqrt{8}}{\sqrt{7} + \sqrt{8}} = \frac{-1(\sqrt{7} + \sqrt{8})}{2(\sqrt{7} - \sqrt{8})(\sqrt{7} + \sqrt{8})}\)
Simplifying the numerator:
\(\frac{(-\sqrt{7} - \sqrt{8})}{2(\sqrt{7} - \sqrt{8})(\sqrt{7} + \sqrt{8})}\)
Using the difference of squares identity again:
\(\sqrt{7} \cdot \sqrt{7} - \sqrt{8} \cdot \sqrt{7} - \sqrt{7} \cdot \sqrt{8} - \sqrt{8} \cdot \sqrt{8}\)
Simplifying further:
\(7 - \sqrt{56} - \sqrt{56} - 8\)
Combining like terms:
\(-1 - 2\sqrt{56}\)
Simplifying the square root:
\(-1 - 2\sqrt{4 \cdot 14}\)
\(-1 - 2 \cdot 2\sqrt{14}\)
\(-1 - 4\sqrt{14}\)
Therefore, the simplified form of the rational number \(\frac{-1}{2(\sqrt{7} - \sqrt{8})}\) is \(-1 - 4\sqrt{14}\).