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.
To determine whether the side length of the square cushion is rational or irrational, we need to consider whether the value of √70 is rational or irrational.
If √70 is rational, then it can be expressed as a fraction p/q, where p and q are integers with no common factors other than 1. However, if √70 is irrational, it cannot be expressed as a fraction and has an infinite number of non-repeating decimal places.
To determine the nature of √70, we can consider its prime factors.
The prime factorization of 70 is 2 * 5 * 7.
Since there is no perfect square that can be multiplied by 2, 5, or 7 to result in 70, we can conclude that √70 is not a rational number.
Therefore, the side length of the square cushion (√70) is irrational.
To determine whether the side length of a square seat cushion is rational or irrational, we need to consider the properties of rational and irrational numbers.
Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. They can be written in the form p/q, where p and q are integers and q is not equal to zero.
Irrational numbers, on the other hand, cannot be written as a fraction and do not terminate or repeat. They are numbers like π (pi) or √2.
In this case, the side length of the square seat cushion is given as 70⎯⎯⎯⎯√ square inches. This notation indicates that the side length is the square root of 70.
Now, we need to determine whether the square root of 70 is rational or irrational. We can do this by considering whether it can be expressed as a fraction.
If the square root of 70 can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero, then it is rational. Otherwise, it is irrational.
To check whether the square root of 70 is rational, we can use prime factorization. Let's find the prime factorization of 70:
70 = 2 × 5 × 7
Since the prime factorization of 70 contains a factor of 7, we know that the square root of 70 cannot be expressed as a fraction of integers. Therefore, the side length of the square seat cushion (√70) is an irrational number.
In conclusion, the side length of the square seat cushion (√70) is irrational because it cannot be expressed as a fraction of integers. The square root of 70 is an irrational number.
To determine whether the side length of a square seat cushion, which is given as 70⎯⎯⎯⎯√ square inches, is rational or irrational, we need to understand the difference between these two types of numbers.
1. Rational Numbers: These are numbers that can be expressed as a fraction, where both the numerator and denominator are integers (whole numbers) and the denominator is not zero. Rational numbers can be shown as terminating decimals or repeating decimals.
2. Irrational Numbers: These are numbers that cannot be expressed as a fraction and do not terminate or repeat when written as decimals. Irrational numbers include the square root of non-perfect squares or any non-repeating/non-terminating decimal.
Now, let's determine whether 70⎯⎯⎯⎯√ is rational or irrational. To do this, we need to simplify the square root expression:
√70 = √(10 * 7)
Since both 10 and 7 are prime numbers, we cannot simplify this further. Therefore, √70 is an irrational number.
Now, let's consider the square of √70:
(√70)^2 = 70
Since 70 is a perfect square, it can be expressed as the fraction 70/1, so it is a rational number.
To conclude, the side length of the square seat cushion is an irrational number (√70), while its area (70^2) is a rational number (70).