A square of a side length 𝑥 has an area of 280 centimetres squared. Which of the following is true about 𝑥? We will be deciding whether it’s an irrational, negative, integer, natural, or rational number.
Before deciding, let’s look at the question. We have a square with side length 𝑥, which means they’re all 𝑥 because in a square all of the sides are equal. And we also know that the area is equal to 280 centimetres squared. The area of the square would be equal to the side squared or length times width. The reason why we can say side squared is because the length and the width are both the same thing. It’s just the side length multiplied to itself or we could write that as the side squared.
So to solve for 𝑥, we can replace area with 𝑥 times 𝑥 equals 280 and 𝑥 times 𝑥 or length times width is equal to 𝑥 squared. So to solve for 𝑥, we need to square root both sides. To break down the square of 280, let’s use a tree — branch method. So 280 is four times 70. Four is a perfect square because it’s two times two, 70 is two times 35, and 35 is five times seven.
So this means that the pair of twos comes on the outside of the square root and all of the numbers on the outside of the branches will go on the inside multiplied together. So two times five times seven is 70. So two squared is 70 or if we would take this into our calculator, we get around 16.43167672 and so on.
So now we need to decide which of the following is true about 𝑥. Let’s begin with A. What is an irrational number? An irrational number is a nonterminating, meaning it never ends, and a nonrepeating decimal — it doesn’t repeat. And it also cannot be written as a fraction. A negative number is exactly what it says: the number is negative, so example, negative five. Integer numbers are our negative numbers, zero, and our positive numbers. But this does not include the numbers between them. Natural numbers are our counting numbers: one, two, three, four, five, all the way up into infinity. And it also does not include the numbers between these. A rational number is a terminating or a repeating decimal. It also could be written as a fraction.
So looking at 𝑥 in a decimal form and square root form, let’s decide which ones we can eliminate. Starting from the bottom, we can eliminate E. Our decimal does not terminate. It doesn’t end nor does it repeat, it’s definitely not a natural number because it’s a decimal. And we can use the same reason for integer. It’s a decimal and it’s not negative.
So let’s make sure A, the irrational number, is the correct answer. Do we have a nonterminating and nonrepeating decimal? Yes, it doesn’t end and it doesn’t repeat. Therefore, this would be an irrational number.