Which statement about the product is true
5.643 x 4.6666666.....(Repeating)
A. The product is irrational
B. The product is neither rational nor irrational
C. The nature of the product cannot be determined
D. The product is rational***
The correct statement is D. The product is rational.
To determine the nature of the product 5.643 x 4.6666666...(repeating), we need to evaluate the expression.
First, let's handle the repeating decimal: 4.6666666...(repeating). This can be expressed as a fraction by setting up an equation where x = 4.6666666...(repeating):
x = 4.6666666...(repeating)
10x = 46.6666666...(repeating)
Subtracting the first equation from the second equation:
10x - x = 46.6666666...(repeating) - 4.6666666...(repeating)
9x = 42
Dividing both sides of the equation by 9:
x = 42/9
x = 4.6666666...(repeating)
So, the repeating decimal 4.6666666...(repeating) can be simplified to the fraction 42/9 or 14/3.
Now, we can multiply 5.643 by 4.6666666...(repeating) or 14/3:
5.643 x 4.6666666...(repeating) = 5.643 x (14/3)
To multiply fractions, we multiply the numerators (5.643 x 14) to get the new numerator and multiply the denominators (1 x 3) to get the new denominator:
(5.643 x 14)/(1 x 3) = 79.002/3
Since 79.002/3 can be expressed as a ratio of two integers, it is a rational number.
Therefore, the correct statement about the product 5.643 x 4.6666666...(repeating) is:
D. The product is rational.