Which statement is true about the relationships between the number sets?(1 point)
Responses
Whole numbers include all natural numbers and 0.
Whole numbers include all natural numbers and 0.
Some irrational numbers are also integers.
Some irrational numbers are also integers.
All rational numbers are also integers.
All rational numbers are also integers.
Not all natural numbers are real numbers.
31 answers
Not all natural numbers are real numbers.
Which of the following is true about −9?(1 point)
Responses
It is both an integer and a whole number.
It is both an integer and a whole number.
It is a whole number but not an integer.
It is a whole number but not an integer.
It is an integer but not a whole number.
It is an integer but not a whole number.
It is an integer but not a rational number.
Responses
It is both an integer and a whole number.
It is both an integer and a whole number.
It is a whole number but not an integer.
It is a whole number but not an integer.
It is an integer but not a whole number.
It is an integer but not a whole number.
It is an integer but not a rational number.
It is both an integer and a whole number.
A sign in a gas station advertises gas at the rate of $3.39910 per gallon. How many dollars does a gallon cost, writing the price strictly as a decimal?(1 point)
$?
$?
The price of gas per gallon is $3.39910.
Which set of numbers is always rational?(1 point)
Responses
decimals
decimals
integers
integers
fractions
fractions
positive numbers
Responses
decimals
decimals
integers
integers
fractions
fractions
positive numbers
integers
The height of an equilateral triangle equals 3√2 times the side length. Using 74 for the square root of 3, what is the approximate height in meters of an equilateral triangle with a side length of 8 meters?(1 point)
The height of an equilateral triangle is given by the formula:
height = side length * √3
Given that the square root of 3 is approximately 1.732, we can substitute it into the formula:
height = 8 * 1.732
Calculating the result:
height = 13.856 meters
Therefore, the approximate height of an equilateral triangle with a side length of 8 meters is 13.856 meters.
height = side length * √3
Given that the square root of 3 is approximately 1.732, we can substitute it into the formula:
height = 8 * 1.732
Calculating the result:
height = 13.856 meters
Therefore, the approximate height of an equilateral triangle with a side length of 8 meters is 13.856 meters.
The diagonal of a square equals 2–√ times the side length. Using 1.4 for the square root of 2, what is the approximate length in feet from one corner to an opposite corner of a square park that is 700 feet on a side?(1 point)
The diagonal of a square is given by the formula:
diagonal = side length * (2 - √2)
Given that the square root of 2 is approximately 1.4, we can substitute it into the formula:
diagonal = 700 * (2 - 1.4)
Calculating the result:
diagonal = 700 * 0.6
diagonal = 420 feet
Therefore, the approximate length from one corner to an opposite corner of the square park that is 700 feet on a side is 420 feet.
diagonal = side length * (2 - √2)
Given that the square root of 2 is approximately 1.4, we can substitute it into the formula:
diagonal = 700 * (2 - 1.4)
Calculating the result:
diagonal = 700 * 0.6
diagonal = 420 feet
Therefore, the approximate length from one corner to an opposite corner of the square park that is 700 feet on a side is 420 feet.
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 point)
40−−√ is between
and
.
40−−√ is between
and
.
To find the whole number and rational number nearest to √40, we can refer to a number line.
The square root of 40 is approximately 6.32. To find the whole number and rational number nearest to √40, we look for the numbers between which √40 falls.
On the number line, 6 is to the left of 6.32 and 7 is to the right. The rational number nearest to √40 is 6.5 (which is halfway between 6 and 7), and the whole number nearest to √40 is 6.
Therefore, 40−−√ is between 6 and 6.5.
The square root of 40 is approximately 6.32. To find the whole number and rational number nearest to √40, we look for the numbers between which √40 falls.
On the number line, 6 is to the left of 6.32 and 7 is to the right. The rational number nearest to √40 is 6.5 (which is halfway between 6 and 7), and the whole number nearest to √40 is 6.
Therefore, 40−−√ is between 6 and 6.5.
Using a number line, what whole number and rational number to the nearest half is 58−−√ between?(1 point)
Responses
between 8 and 8.5
between 8 and 8.5
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
Responses
between 8 and 8.5
between 8 and 8.5
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
To find the whole number and rational number nearest to √58, we can refer to a number line.
The square root of 58 is approximately 7.62. To find the whole number and rational number nearest to √58, we look for the numbers between which √58 falls.
On the number line, 7 is to the left of 7.62 and 8 is to the right. The rational number nearest to √58 is 7.5 (which is halfway between 7 and 8), and the whole number nearest to √58 is 7.
Therefore, 58−−√ is between 7 and 7.5.
The square root of 58 is approximately 7.62. To find the whole number and rational number nearest to √58, we look for the numbers between which √58 falls.
On the number line, 7 is to the left of 7.62 and 8 is to the right. The rational number nearest to √58 is 7.5 (which is halfway between 7 and 8), and the whole number nearest to √58 is 7.
Therefore, 58−−√ is between 7 and 7.5.
Using rational approximations, what is an inequality statement comparing 81−−√ and 8–√8? Enter <, >, or =.(1 point)
The inequality is: 81−−√
8–√8.
The inequality is: 81−−√
8–√8.
To compare 81−−√ and 8–√8 using rational approximations, we can simplify each expression.
81−−√ is equal to 9, since the square root of 81 is 9.
8–√8 can be simplified as follows:
√8 is approximately 2.828
8–2.828 is approximately 5.172
So, the inequality statement comparing 81−−√ and 8–√8 is:
9 > 5.172
81−−√ is equal to 9, since the square root of 81 is 9.
8–√8 can be simplified as follows:
√8 is approximately 2.828
8–2.828 is approximately 5.172
So, the inequality statement comparing 81−−√ and 8–√8 is:
9 > 5.172
Write 6,739 in expanded form.(1 point)
Responses
(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
(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
(7×100)+(3×10)+(9×1)
left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 9 times 1 right parenthesis
(9×1,000)+(3×100)+(7×10)+(6×1)
Responses
(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
(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
(7×100)+(3×10)+(9×1)
left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 9 times 1 right parenthesis
(9×1,000)+(3×100)+(7×10)+(6×1)
(6×1,000)+(7×100)+(3×10)+(9×1)
or
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
or
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
Express 811 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
To express 811 as a repeating decimal, we divide 811 by 99, which is the largest double-digit number that evenly divides into 811.
The division is as follows:
811 ÷ 99 = 8.20202020...
In this case, the digit "02" repeats indefinitely. Therefore, we express 811 as a repeating decimal by marking the cycle of repeating digits as:
8.20¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Note: The line over the "20" indicates that "02" repeats infinitely.
The division is as follows:
811 ÷ 99 = 8.20202020...
In this case, the digit "02" repeats indefinitely. Therefore, we express 811 as a repeating decimal by marking the cycle of repeating digits as:
8.20¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Note: The line over the "20" indicates that "02" repeats infinitely.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.(1 point)
To convert the repeating decimal 0.72¯¯¯¯¯ to a rational number in simplest form, we can use algebraic manipulation.
Let x = 0.72¯¯¯¯¯
Multiplying both sides of the equation by 1000 to eliminate the repeating decimal:
1000x = 727.272727...
Subtracting x from both sides:
1000x - x = 727.272727... - 0.72¯¯¯¯¯
999x = 726.552727...
To eliminate the decimal point and repeating decimal, we can multiply by a power of 10. In this case, multiplying by 10000 will suffice:
999x × 10000 = 7265527.272727...
10000x = 7265527.272727...
Now we can simplify the equation to find x:
x = 7265527.272727... ÷ 10000
x = 726.552727...
Since 726.552727... is a fraction, we can express it in simplest form by writing it as a ratio of two integers:
x = 726.552727... = 726 + 0.552727...
= 726 + 552727/999999
Simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor:
x = 726 + 552727/999999
x = 726 + 77/141
Therefore, the rational number in simplest form equivalent to the repeating decimal 0.72¯¯¯¯¯ is 726 + 77/141.
Let x = 0.72¯¯¯¯¯
Multiplying both sides of the equation by 1000 to eliminate the repeating decimal:
1000x = 727.272727...
Subtracting x from both sides:
1000x - x = 727.272727... - 0.72¯¯¯¯¯
999x = 726.552727...
To eliminate the decimal point and repeating decimal, we can multiply by a power of 10. In this case, multiplying by 10000 will suffice:
999x × 10000 = 7265527.272727...
10000x = 7265527.272727...
Now we can simplify the equation to find x:
x = 7265527.272727... ÷ 10000
x = 726.552727...
Since 726.552727... is a fraction, we can express it in simplest form by writing it as a ratio of two integers:
x = 726.552727... = 726 + 0.552727...
= 726 + 552727/999999
Simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor:
x = 726 + 552727/999999
x = 726 + 77/141
Therefore, the rational number in simplest form equivalent to the repeating decimal 0.72¯¯¯¯¯ is 726 + 77/141.
Which of the following is a rational number between 5 and 514?(1 point)
Responses
26−−√
start root 26 end root
513
5 Start Fraction 1 over 3 end fraction
525
5 Start Fraction 2 over 5 end fraction
517
Responses
26−−√
start root 26 end root
513
5 Start Fraction 1 over 3 end fraction
525
5 Start Fraction 2 over 5 end fraction
517
The rational number between 5 and 514 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.(1 point)
$$
$$
To find a rational number with a denominator of 7 that is between 7–√ and 8–√, we can consider the decimal expansions of these numbers.
7–√ is approximately 5.64, and 8–√ is approximately 6.35. Between these two decimals, we are looking for a rational number with a denominator of 7.
The closest rational number with a denominator of 7 between these two decimals is \(\frac{5}{7}\).
Therefore, a rational number with a denominator of 7 between 7–√ and 8–√ is \(\frac{5}{7}\).
7–√ is approximately 5.64, and 8–√ is approximately 6.35. Between these two decimals, we are looking for a rational number with a denominator of 7.
The closest rational number with a denominator of 7 between these two decimals is \(\frac{5}{7}\).
Therefore, a rational number with a denominator of 7 between 7–√ and 8–√ is \(\frac{5}{7}\).
What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(1 point)
$$????
$$????
To find a rational number with a denominator of 7 that is between 7−√ and 8−√, we can add the two radicals and divide the sum by 2:
(7−√ + 8−√ ) / 2
Simplifying the expression:
(15−√ ) / 2
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is 2+√:
(15−√) / 2 * (2+√) / (2+√)
Expanding the numerator and denominator:
(30 + 15√ − 2√ − √²) / (4 − √²)
Combining like terms:
(30 + 13√ − √²) / (4 − √²)
Since √² is equal to 2, we can simplify further:
(30 + 13√ − 2) / (4 − 2)
Simplifying:
(28 + 13√) / 2
Therefore, a rational number with a denominator of 7 that is between 7−√ and 8−√ is (28 + 13√) / 2.
(7−√ + 8−√ ) / 2
Simplifying the expression:
(15−√ ) / 2
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is 2+√:
(15−√) / 2 * (2+√) / (2+√)
Expanding the numerator and denominator:
(30 + 15√ − 2√ − √²) / (4 − √²)
Combining like terms:
(30 + 13√ − √²) / (4 − √²)
Since √² is equal to 2, we can simplify further:
(30 + 13√ − 2) / (4 − 2)
Simplifying:
(28 + 13√) / 2
Therefore, a rational number with a denominator of 7 that is between 7−√ and 8−√ is (28 + 13√) / 2.
Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer.(4 points)
The relationship between rational numbers and whole numbers is as follows:
1. All whole numbers are rational numbers: Whole numbers are a subset of rational numbers. A rational number is defined as any number that can be expressed as a ratio of two integers, where the denominator is not zero. Since all whole numbers can be expressed as a ratio of an integer divided by 1, they are considered rational numbers. For example, 5 is a whole number and can be expressed as the ratio 5/1, which is a rational number.
2. Not all rational numbers are whole numbers: While all whole numbers are rational, not all rational numbers are whole numbers. Rational numbers include numbers that can be expressed as fractions, terminating decimals, recurring decimals, or integers. Examples of rational numbers that are not whole numbers are 3/5, -1/2, and 4.25.
In conclusion, all whole numbers are rational numbers, but not all rational numbers are whole numbers.
1. All whole numbers are rational numbers: Whole numbers are a subset of rational numbers. A rational number is defined as any number that can be expressed as a ratio of two integers, where the denominator is not zero. Since all whole numbers can be expressed as a ratio of an integer divided by 1, they are considered rational numbers. For example, 5 is a whole number and can be expressed as the ratio 5/1, which is a rational number.
2. Not all rational numbers are whole numbers: While all whole numbers are rational, not all rational numbers are whole numbers. Rational numbers include numbers that can be expressed as fractions, terminating decimals, recurring decimals, or integers. Examples of rational numbers that are not whole numbers are 3/5, -1/2, and 4.25.
In conclusion, all whole numbers are rational numbers, but not all rational numbers are whole numbers.