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The side length of a square seat cushion is 70⎯⎯⎯⎯√
square inches. Determine whether the side length is rational or irrational. Then explain your reasoning.
square inches. Determine whether the side length is rational or irrational. Then explain your reasoning.
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GPT 3.5
The side length of a square seat cushion is expressed as 70√2 square inches.
To determine whether this side length is rational or irrational, we need to determine if √2 is rational or irrational.
Assume √2 is rational, meaning it can be expressed as a fraction in the form a/b, where a and b are integers and b is not zero.
If √2 is rational, we can express it as (a/b) = √2. By squaring both sides of the equation, we get 2 = (a^2/b^2). Multiplying both sides by b^2, we have 2b^2 = a^2.
From this equation, we can see that a^2 must be even since it is equal to 2b^2. This implies that a must also be even (since the square of an odd number is odd).
If a is even, we can express it as a = 2c, where c is an integer. Substituting this expression into 2b^2 = a^2, we have 2b^2 = (2c)^2, which simplifies to b^2 = 2c^2.
From this equation, we can also see that b^2 must be even since it is equal to 2c^2. This implies that b must also be even.
Thus, both a and b must be even, which means there is a common factor of 2 for both a and b. However, this contradicts our assumption that a/b is in its simplest form, because if a and b have a common factor of 2, they can be further reduced.
Therefore, our assumption that √2 is rational is incorrect. Thus, √2 is irrational.
Since the side length of the square cushion is expressed as 70√2, we can conclude that the side length is irrational.
To determine whether this side length is rational or irrational, we need to determine if √2 is rational or irrational.
Assume √2 is rational, meaning it can be expressed as a fraction in the form a/b, where a and b are integers and b is not zero.
If √2 is rational, we can express it as (a/b) = √2. By squaring both sides of the equation, we get 2 = (a^2/b^2). Multiplying both sides by b^2, we have 2b^2 = a^2.
From this equation, we can see that a^2 must be even since it is equal to 2b^2. This implies that a must also be even (since the square of an odd number is odd).
If a is even, we can express it as a = 2c, where c is an integer. Substituting this expression into 2b^2 = a^2, we have 2b^2 = (2c)^2, which simplifies to b^2 = 2c^2.
From this equation, we can also see that b^2 must be even since it is equal to 2c^2. This implies that b must also be even.
Thus, both a and b must be even, which means there is a common factor of 2 for both a and b. However, this contradicts our assumption that a/b is in its simplest form, because if a and b have a common factor of 2, they can be further reduced.
Therefore, our assumption that √2 is rational is incorrect. Thus, √2 is irrational.
Since the side length of the square cushion is expressed as 70√2, we can conclude that the side length is irrational.
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