Which of the following is a rational number?(1 point)
Responses
88√
the fraction with numerator 8 and denominator square root of 8
π
pi
7–√
the fraction with numerator 8 and denominator square root of 8
Which of the following is a rational number?(1 point)
Responses
π
pi
88√
the fraction with numerator 8 and denominator square root of 8
7–√
square root of 7
7.8
To determine if a number is rational, we need to see if it can be expressed as a fraction of two integers.
Let's evaluate each option:
1. 88√: This is an irrational number because the square root of any non-perfect square is irrational.
2. The fraction with numerator 8 and denominator square root of 8: To determine if this fraction is rational, we need to simplify it. The square root of 8 is √8 = 2√2. Therefore, the fraction can be simplified to 8/(2√2), which can further be simplified to 4/√2. To rationalize the denominator, we multiply numerator and denominator by the conjugate of √2, which is also √2. This gives us (4/√2)(√2/√2) = 4√2/2 = 2√2. Since this fraction can be expressed as a ratio of two integers (numerator 2 and denominator 1), it is a rational number.
3. π (pi): Pi is an irrational number and cannot be expressed as a fraction of two integers.
4. 7–√: This is an irrational number as the square root of any non-perfect square is irrational.
Therefore, the only rational number among the options is the fraction with numerator 8 and denominator square root of 8.
To determine which of the given options is a rational number, we need to understand what a rational number is.
A rational number is any number that can be expressed as a fraction, where both the numerator and the denominator are integers.
Let's analyze each option one by one:
1) 88√: This is an irrational number because it involves the square root (√) of a non-perfect square. Irrational numbers cannot be expressed as fractions.
2) The fraction with numerator 8 and denominator square root of 8: To determine if this is a rational number, we need to simplify the fraction. The square root of 8 can be simplified to 2√2. So, the fraction becomes 8 / (2√2). To make it rational, we can rationalize the denominator by multiplying both the numerator and the denominator by √2. This gives us 8√2 / (2√2 * √2), which simplifies to (8√2) / (2 * 2), and further simplifies to 4√2. Since 4√2 cannot be expressed as a fraction, this option is not a rational number.
3) π (pi): This is an irrational number because it represents the mathematical constant pi, which is the ratio of a circle's circumference to its diameter. Irrational numbers cannot be expressed as fractions.
4) 7–√ : This is an irrational number because it involves the square root (√) of 7, which is a non-perfect square. Irrational numbers cannot be expressed as fractions.
Therefore, the only rational number among the given options is none of the above.