Question

PLEASE HELP ME

pick one of the answers below to explain why 0, point, 7, 4, 7, 4, 4, 7, 4, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 4, 4, 7, point, point, .0.747447444744447444447... . is a rational number:
A It is a decimal that does terminate
B it is the square root of a perfect square
C it is a square root of a non-perfect square

Answer 1

[AI-generated answer — Bot GPT 3.5]

C It is a square root of a non-perfect square

A rational number is one that can be expressed as the ratio of two integers. The given decimal is a repeating decimal, which means it can be expressed as a fraction.

To show that it is a square root of a non-perfect square, we first need to realize that the decimal 0.747447444744447444447... can be written as 7/10 + 4/100 + 74/1000 + 744/10000 + 7(4444...)/10000000... = 7/10 + 4/100 + 74/1000 + 744/10000 + 7/(10^n) * (4444...)/(10000...), where n represents the number of 4's after the decimal point.

Now, let's simplify this expression as a fraction. We can rewrite it as (7/10 + 4/100 + 74/1000 + 744/10000)/(1 - 1/10^n).

The numerator is a finite sum of fractions, which is a rational number. The denominator, 1 - 1/10^n, is also a rational number because it is the difference of two rational numbers. Therefore, the entire expression is a rational number.

Hence, the given decimal is a rational number.

Answer 2

[AI-generated answer — Step-by-Step Bot]

The correct answer is C: It is a square root of a non-perfect square.

To explain why, we need to understand what a rational number is. A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.

In this case, the number 0.747447444744447444447... is a repeating decimal, often denoted as an infinite repeating block of digits. To determine whether it is rational, we need to express it as a fraction.

Let's call the repeating block of digits "x". So we have:

0.747447444744447444447... = 0.x

To get rid of the repeating part, we can subtract "x" from the original number:

10x = 7.47447444744447444447...

Now we subtract the original equation from the new equation:

10x - x = 7.47447444744447444447... - 0.x

This simplifies to:

9x = 7.47447444744447444447...

To express this as a fraction, we move the decimal point to keep only the repeating block:

9x = 747.444744444744444...

Now we can see that x = 747.444744444744444... / 9.

To simplify this fraction, we divide the numerator and denominator by their greatest common divisor, which is 3:

x = 747.444744444744444... / 9 = 249.148248148148148... / 3

Therefore, we can express the number 0.747447444744447444447... as a rational number:

x = 249.148248148148148... / 3

Since it can be expressed as the quotient of two integers (249.148248148148148 and 3), where the denominator is not zero, it is a rational number.

Answer 3

[AI-generated answer — Explain Bot]

To determine whether 0.747447444744447444447... is a rational number, we need to understand the definitions of rational and irrational numbers.

A rational number is a number that can be expressed as the ratio of two integers, where the denominator is not zero. It can be written as either a terminating decimal or a repeating decimal.

An irrational number, on the other hand, cannot be expressed as the ratio of two integers. It cannot be written as a terminating or repeating decimal.

Now, let's analyze the given number 0.747447444744447444447... and discuss each option:

A) It is a decimal that does terminate: This statement is not applicable to the given number because the decimal representation of 0.747447444744447444447... does not terminate. It continues infinitely with repeating patterns.

B) It is the square root of a perfect square: This is also not true because the given number does not represent a square root of any perfect square. In a perfect square, the decimal representation would either terminate or repeat in a specific pattern.

C) It is a square root of a non-perfect square: This option is also not valid as the given number does not represent the square root of any non-perfect square. The decimal representation of the square root of a non-perfect square would not repeat in a pattern like 0.747447444744447444447... does.

Therefore, none of the options (A, B, or C) accurately explain why 0.747447444744447444447... is a rational number. In fact, this number is an example of an irrational number since it cannot be written as a ratio of two integers and has a pattern of repeating decimals.

To explain why this number is rational, we need to consider other possibilities or statements that can accurately describe its properties.