select the answer with the rational product and a correct explanation for why the product is it rational
(0.333…)•14/17 because the product of a repeating decimal in a rational number is always a rational number
v3•v27 because the product of two irrational numbers is always a rational number
1/4•v16 because the product of a fraction, and a square root is always a rational number
2/pi•pi because the product of a fraction in an irrational number is always a rational number
1/4•v16 because the product of a fraction and a square root of a perfect square is always a rational number. v16 equals 4, which is a perfect square, so the product of 1/4 and v16 is rational.
The correct answer is: (1/4)•√16 because the product of a fraction and a square root is always a rational number.
Explanation:
A rational number can be defined as any number that can be expressed as a fraction of two integers. In the given expression, (1/4) is a fraction, and √16 is a square root of a perfect square.
The square root of 16 is 4, which is a whole number and can be expressed as a fraction (4/1).
So, when we multiply (1/4), which is a rational number, by √16, which is also a rational number since it can be expressed as a fraction (4/1), the product will be a rational number.
Therefore, the product (1/4)•√16 is rational.
The correct answer is (0.333...) • 14/17 because the product of a repeating decimal and a rational number is always a rational number.
To understand why this is the case, we need to understand the definition of a rational number and how multiplication works.
A rational number is any number that can be expressed as a ratio of two integers, where the denominator is not zero. In other words, it can be written as p/q, where p and q are integers and q is not equal to zero.
Now, let's analyze the product (0.333...) • (14/17). The repeating decimal 0.333... can be represented as 1/3, which is a rational number. Multiplying a rational number with another rational number will always result in a rational number.
To compute the product, we convert 0.333... to 1/3:
(0.333...) • (14/17) = (1/3) • (14/17)
Now, we multiply the numerators together, and multiply the denominators together:
(1/3) • (14/17) = (1 • 14) / (3 • 17) = 14/51
As we can see, the product 14/51 is a rational number since it can be expressed as the ratio of two integers, where the denominator is not zero.
Therefore, the product (0.333...) • (14/17) is a rational number, making it the correct answer.