Question
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 between 7–√ and 8–√ with a denominator of 7, we can average the two given numbers.
7 – √ = 7 – √/1 = 7/1 – √/1
8 – √ = 8 – √/1 = 8/1 – √/1
To average two fractions, we need to have the same denominator. So to find a common denominator, we can multiply the denominators together:
Denominator = 1 × 1 = 1
Now, let's calculate the average:
(7/1 – √/1 + 8/1 – √/1) / 2 =
(7 – √ + 8 – √) / 2 =
15 – 2√ / 2 =
(15/2 – √) / 1 =
15/2 – √/1
Therefore, a rational number between 7–√ and 8–√ with a denominator of 7 is 15/2 – √.
Please explain what 15/2 – √, I don't get it
I apologize for the confusion. Let's simplify and rephrase the answer.
To find a rational number between 7–√ and 8–√ with a denominator of 7, we can average the two given numbers.
The given numbers are:
7 – √ and 8 – √
To average two numbers, we add them together and divide by 2:
(7 – √ + 8 – √) / 2
Now, let's simplify:
7 – √ + 8 – √
= 15 – 2√
Since we want the answer as an improper fraction, we can express it as:
(15 – 2√) / 1
Therefore, a rational number between 7–√ and 8–√ with a denominator of 7 is (15 – 2√) / 1.
I hope this clears up any confusion. Let me know if there's anything else you would like clarification on!
What does (15 – 2√) / 1 mean. Please give an answer like 1/7: one-seventh
Apologies for the confusion created in my previous response. Let's simplify the expression.
To simplify (15 – 2√) / 1, we can write it as a fraction by dividing the numerator by the denominator:
(15 – 2√) / 1 = (15 – 2√)
Now, to express it as an improper fraction, we need to find a common denominator. Since the expression does not have a denominator other than 1, the format in the question is already in the correct form.
Therefore, the irrational number (15 – 2√) / 1 does not need to be converted to an improper fraction. It is best expressed as (15 – 2√).
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.
To convert the recurring decimal 0.72¯¯¯¯¯ to a rational number in simplest form, we can use a few algebraic steps.
Let's assign a variable to the recurring part, denoted as x:
0.72¯¯¯¯¯ = 0.7 + x
Since the recurring part extends infinitely, multiplying both sides of the equation by 10 eliminates the decimal point in the recurring part:
10(0.72¯¯¯¯¯) = 10(0.7 + x)
This simplifies to:
7.2¯¯¯¯¯ = 7 + 10x
Next, we subtract the original equation from the above equation to eliminate the recurring part:
10(0.72¯¯¯¯¯) - 0.72¯¯¯¯¯ = 7 + 10x - (0.7 + x)
9.72 - 0.72¯¯¯¯¯ = 7 + 9x
9 = 7 + 9x
Now we can solve for x:
9 - 7 = 9x
2 = 9x
x = 2/9
Therefore, the rational number equivalent to 0.72¯¯¯¯¯ in simplest form is 2/9.
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.
The relationship between rational numbers and whole numbers is that every whole number is a rational number, but not every rational number is a whole number.
A rational number is a number that can be expressed as a fraction, where both the numerator and denominator are integers. Whole numbers are a subset of rational numbers, where the denominator is always 1.
For example, the number 3 can be expressed as a fraction 3/1, making it both a whole number and a rational number.
However, not all rational numbers are whole numbers. For example, the number 2.5 is a rational number because it can be expressed as the fraction 5/2. However, it cannot be classified as a whole number because it is not an integer.
In conclusion, while all whole numbers are rational numbers, not all rational numbers are whole numbers.
To find a rational number with a denominator of 7 between 7–√ and 8–√, we need to rationalize the denominators of the given numbers.
1. Let's start with 7–√. To rationalize the denominator, we multiply the expression by its conjugate, which is 7+√.
(7–√) * (7+√) = (7^2) – (√)^2 = 49 – √49 = 49 – 7 = 42.
2. Now, let's rationalize the denominator of 8–√. We multiply the expression by its conjugate, which is 8+√.
(8–√) * (8+√) = (8^2) – (√)^2 = 64 – √64 = 64 – 8 = 56.
So, the rational numbers we obtained are 42/7 and 56/7.
Since we need a value between these two numbers, we take the average of these fractions.
(42/7 + 56/7) / 2 = 98/14 = 7.
Therefore, the rational number with a denominator of 7 between 7–√ and 8–√ is 7, which can be written as the improper fraction 7/1.