Which statement about the product is true?
7.56 *6. overline 7
(1 point)
The product is irrational.
The product is rational.
O The product is neither rational nor irrational.
The nature of the product cannot be determined.
The product is irrational.
To determine whether the product 7.56 * 6.7777... is rational or irrational, we need to analyze the nature of the repeating decimal part, which is '7777...'.
A rational number can be written as a fraction of two integers, while an irrational number cannot be expressed as a fraction.
In this case, since the decimal part repeats indefinitely, we can express it as a fraction: 7.56 * 6.7777... = (756/100) * (6777.../1000).
Thus, the product is rational.
Therefore, the correct statement is: The product is rational.
To determine whether the product 7.56 * 6. overline 7 is rational or irrational, we need to understand the concept of rational and irrational numbers.
A rational number is any number that can be written as a fraction or a ratio of two integers (whole numbers), where the denominator is not zero. Rational numbers can be expressed as terminating decimals (e.g., 0.25, 0.75) or repeating decimals (e.g., 0.333..., 0.121212...).
An irrational number, on the other hand, is a number that cannot be expressed as a fraction or a ratio of two integers. Irrational numbers have non-repeating, non-terminating decimals, such as pi (π) or the square root of 2 (√2).
To determine whether 7.56 * 6. overline 7 is rational or irrational, we can convert it to decimal form and see if it repeats or terminates.
First, let's convert 6. overline 7 to decimal form.
To do this, we can create an equation where x equals 6. overline 7:
x = 0.677777...
Next, we can multiply both sides of the equation by 10 to move the decimal point:
10x = 6.77777...
Now, we subtract the original equation from the new equation:
10x - x = 6.77777... - 0.677777...
This simplifies to:
9x = 6
Now, we solve for x:
x = 6/9 = 2/3 = 0.66666...
Therefore, 6. overline 7 is equivalent to 0.66666...
Next, we can calculate the product of 7.56 and 0.66666... to see if it is rational or irrational:
7.56 * 0.66666... = 5.04
Since 5.04 is a terminating decimal, it can be expressed as a fraction (in this case, 5.04 can be written as 504/100), which means it is a rational number.
Based on the above calculations, the correct statement about the product 7.56 * 6. overline 7 is:
The product is rational.