Which statement about the product is true? 6.421 x 2 .̅6
-The product is rational.
-The product is irrational.
-The product is neither rational nor irrational.
-The nature of the product cannot be determined.
The product is irrational.
To determine whether the product 6.421 x 2 .̅6 is rational, irrational or if its nature cannot be determined, we need to understand the definitions of rational and irrational numbers.
A rational number is any number that can be expressed as a fraction, where both the numerator and the denominator are integers.
An irrational number is a number that cannot be expressed as a fraction and its decimal representation goes on infinitely without repeating.
Now, let's calculate the product 6.421 x 2 .̅6.
Step 1: Convert the repeating decimal to a fraction.
Let x = 0.666...
then 10x = 6.666...
Now, subtract the first equation from the second equation:
10x - x = 6.666... - 0.666...
Simplifying:
9x = 6
Divide both sides by 9:
x = 6/9 = 2/3
Step 2: Calculate the product 6.421 x 2/3.
6.421 x 2/3 = (6.421)x(2/3)
= 12.842/3
= 4.281333...
Step 3: Analyze the result.
The product 6.421 x 2 .̅6 is approximately 4.281333...
Since this number can be expressed as a fraction (2/3) and is not a repeating or non-repeating decimal, it can be considered rational.
Therefore, a true statement about the product is that it is rational.