Using a number line, what whole number and rational number to the nearest half is 58−−√ between?
Responses
between 8.5 and 9
between 8.5 and 9
between 7 and 7.5
between 7 and 7.5
between 7.5 and 8
between 7.5 and 8
between 8 and 8.5
between 8 and 8.5
Using rational approximations, what is an inequality statement comparing 81−−√ and 8–√8? Enter <, >, or =.(1 point)
The inequality is: √81 –√88
The inequality is: √81 < √88
Write 6,739 in expanded form.(1 point)
Responses
(9×1,000)+(3×100)+(7×10)+(6×1)
left parenthesis 9 times 1,000 right parenthesis plus left parenthesis 3 times 100 right parenthesis plus left parenthesis 7 times 10 right parenthesis plus left parenthesis 6 times 1 right parenthesis
(6×1,000)+(7×100)+(3×10)+(9×1)
left parenthesis 6 times 1,000 right parenthesis plus left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 9 times 1 right parenthesis
(6×1,000)+(7×100)+(3×10)+(0×1)
left parenthesis 6 times 1,000 right parenthesis plus left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 0 times 1 right parenthesis
(7×100)+(3×10)+(9×1)
(6×1,000)+(7×100)+(3×10)+(9×1)
Express 811 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
811 as a repeating decimal is 811.
Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
To express 8/11 as a repeating decimal, we can perform long division:
72
___________
11 | 8.000
7 7
________
300
286
________
140
132
________
80
77
________
30
22
________
80
The remainder is 80. This means that the decimal representation of 8/11 is approximately 0.727272... The digits 72 repeat indefinitely, so we can write it as 0.72¯¯¯.
explain
To express 8/11 as a repeating decimal, we use long division to divide 8 by 11. The quotient is 0.72 and the remainder is 80. Since the remainder is not zero, we continue dividing by 11.
We bring down a zero from the dividend to get 800. We then divide 800 by 11 to get a quotient of 72 and a remainder of 8. Again, we have a non-zero remainder, so we bring down another zero and continue dividing.
We repeat this process indefinitely, as there is no limit to the number of decimal places we can calculate. The pattern of remainder digits (80, 8) repeats, so we can represent the repeating part with a bar over the repeating digits.
Therefore, we can express 8/11 as the repeating decimal 0.72¯¯¯.
where you get 0.72
Apologies for the error in my previous response. Let's go through the calculation again:
When we divide 8 by 11, the quotient is 0 and the remainder is 8. Since the remainder is smaller than the divisor, we bring down a zero to get 80. Now, we divide 80 by 11.
In the next step, we find that 11 can go into 80 seven times, with a remainder of 3. Again, the remainder is smaller than the divisor, so we bring down a zero to get 30. We divide 30 by 11.
This time, we find that 11 can go into 30 two times, with a remainder of 8. The remainder is still smaller than the divisor, so we bring down another zero to get 80. We divide 80 by 11.
Now, we find that 11 can go into 80 seven times, with a remainder of 3. We bring down another zero, but since we already had a remainder of 3 before, we know that the division will continue in the same pattern.
Thus, we can express 8/11 as the repeating decimal 0.72¯¯¯, where the digits 72 repeat indefinitely. I apologize for the confusion caused by my previous incorrect response.
shorten it
Apologies for the confusion. Upon reviewing the division, 8 divided by 11 is approximately 0.727272... The digits 72 repeat indefinitely, so we can write it as 0.72¯¯¯.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.(1 point)
To convert 0.72¯¯¯¯¯ to a rational number in simplest form, we can set it as a variable "x" and subtract it from the original decimal:
x = 0.727272...
10x = 7.272727...
Subtracting the above equations:
10x - x = 7.272727... - 0.727272...
9x = 6.545454...
Dividing both sides by 9:
x = 0.727272... / 9
Simplifying further:
x = 8/11
Therefore, 0.72¯¯¯¯¯ is equal to the rational number 8/11 in simplest form.
The fraction equivalent of 0.8¯¯¯ is
To find the fraction equivalent of 0.8¯¯¯, we can set it as the variable "x" and subtract it from the original decimal:
x = 0.8¯¯¯
10x = 8.8¯¯¯
Subtracting the above equations:
10x - x = 8.8¯¯¯ - 0.8¯¯¯
9x = 8
Dividing both sides by 9:
x = 8/9
Therefore, the fraction equivalent of 0.8¯¯¯ is 8/9.