Which of the following rational numbers can be expressed as a repeating decimal: 78
, 512
, 1725
, or 3140
?
A rational number can be expressed as a repeating decimal if and only if its decimal representation is non-terminating, meaning that it goes on forever without repeating.
We can determine this by finding the prime factorization of each number and looking at the denominators of the fractions in lowest terms:
78 = 2 x 3 x 13
512 = 2^9
1725 = 3 x 5 x 5 x 23
3140 = 2^2 x 5 x 157
From the prime factorizations, we can see that 78, 1725, and 3140 all have prime factors other than 2 and 5, so their decimal representations will not terminate and will repeat.
512, on the other hand, only has 2 as a prime factor, so its decimal representation will terminate and not repeat.
Therefore, the rational numbers that can be expressed as a repeating decimal are: 78, 1725, and 3140.
Express 713
as a repeating decimal. Mark the repeating digits with a bar.
Note: You will find the option to add a bar above text in the Geometry (∠
) keyboard.
To convert 713 into a repeating decimal, we need to divide 713 by an appropriate power of 10.
Let's start by dividing 713 by 9, which is the largest one-digit number:
713 ÷ 9 = 79 with a remainder of 2
We can rewrite this as:
79 + 2/9
Now we divide the remainder (2) by 9:
2 ÷ 9 = 0.2222...
Putting it all together, we have:
713 = 79.2222...
Thus, the repeating decimal representation of 713 is 79.2¯¯¯¯¯¯.
Convert 0.8¯¯¯
to a rational number
To convert the repeating decimal 0.8¯¯¯¯ to a rational number, we can use algebraic manipulation.
Let x = 0.8¯¯¯¯
Then, we can multiply x by 10 to move the decimal point:
10x = 8.8¯¯¯¯
Now, let's subtract the original equation from the new equation:
10x - x = 8.8¯¯¯¯ - 0.8¯¯¯¯
9x = 8
Dividing both sides by 9, we find:
x = 8/9
So, the rational number equivalent of 0.8¯¯¯¯ is 8/9.
To convert 0.82¯¯¯¯¯
to a rational number, you would set x
equal to 0.82¯¯¯¯¯
and then multiply both sides of the equation by what number?(1 point)
To convert the repeating decimal 0.82¯¯¯¯ to a rational number, we can use algebraic manipulation.
Let x = 0.82¯¯¯¯¯
To remove the repeating decimal, we can multiply both sides of the equation by a power of 10 equal to the number of repeating digits. In this case, there is only one repeating digit (2), so we can multiply both sides by 10:
10x = 8.2¯¯¯¯¯
Now, let's subtract the original equation from the new equation:
10x - x = 8.2¯¯¯¯¯ - 0.82¯¯¯¯¯
Simplifying, we get:
9x = 8.2
Therefore, to convert 0.82¯¯¯¯ to a rational number, we would multiply both sides of the equation by 9.