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.
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 the side length of a square seat cushion is rational or irrational, we need to consider the square root of the side length, which is √70.
First, let's determine if √70 is rational or irrational.
We know that a rational number can be expressed as a fraction in the form p/q, where p and q are integers and q ≠ 0. An irrational number, on the other hand, cannot be expressed as a fraction.
To determine if √70 is rational, let's assume it is rational:
√70 = p/q, where p and q are integers and q ≠ 0.
Now, we can square both sides of the equation to eliminate the square root:
(√70)^2 = (p/q)^2
70 = p^2 / q^2
Rearranging the equation gives us:
70q^2 = p^2
This means that p^2 is a multiple of 70. Since the prime factorization of 70 is 2 * 5 * 7, p^2 must also have these factors.
Now let's observe the prime factorization of p^2. Since p^2 has the same factors as 70, it must also have the factors 2, 5, and 7.
However, if p^2 has these prime factors, then p itself must also have these same prime factors. This implies that p is a multiple of 2, 5, and 7.
Now, if p is a multiple of 2, 5, and 7, then p^2 is also a multiple of 2, 5, and 7.
But we know that 70q^2 = p^2, which means that 70q^2 is also a multiple of 2, 5, and 7.
If 70q^2 is a multiple of 2, 5, and 7, then q^2 must also be a multiple of 2, 5, and 7.
Now let's consider the value of q. Since q^2 is a multiple of 2, 5, and 7, q must also be a multiple of 2, 5, and 7.
However, this contradicts our assumption that p and q have no common factors other than 1. If q is a multiple of 2, 5, and 7, then p/q is not in its simplest form, which is a contradiction.
Therefore, our assumption that √70 is rational must be false. Hence, √70 is irrational.
In conclusion, the side length of the square seat cushion, which is √70 square inches, is an irrational number.
To determine whether the side length of a square seat cushion is rational or irrational, we need to understand the difference between these two types of numbers.
Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero. They can be written in the form of fractions. For example, 1/4, 3/2, and -5/7 are rational numbers.
Irrational numbers, on the other hand, cannot be expressed as a fraction or a ratio of two integers. They are non-repeating, non-terminating decimals. Some examples of irrational numbers include √2, π (pi), and e.
In this case, the side length of the square seat cushion is given as 70⎯⎯⎯⎯√ square inches. To determine whether it is rational or irrational, we can simplify the square root if possible.
Simplifying 70⎯⎯⎯⎯√ :
√70^2 = √4900 = 70
Since 70 is a whole number and can be expressed as the ratio 70/1, the side length of the square seat cushion is rational.
By performing the simplification, we found that the square root of 4900 is a whole number, indicating that the side length can be expressed as a ratio of two integers without a remainder. Therefore, we can conclude that the side length of the square seat cushion is rational.