Consider the square root of 8+x
a) find a value for x that would make this number rational, and explain
b) find a value for x that would make this number irrational, and explain
a) To make the number rational, we need to find a value for x such that the expression can be written as a fraction. A rational number is any number that can be expressed as the ratio of two integers.
If the square root of 8+x is rational, then we can write it as a fraction in the form of sqrt(8+x) = a/b, where a and b are integers, and b is not equal to zero.
To make this possible, we can square both sides of the equation:
(sqrt(8+x))^2 = (a/b)^2
8 + x = a^2/b^2
To solve for x, we can rearrange the equation:
x = a^2/b^2 - 8
By choosing any values for a and b that result in a perfect square when squared and their ratio is not equal to 0, we can make the square root of 8+x rational. For example, let's consider a = 4 and b = 2:
x = (4^2)/(2^2) - 8
x = 16/4 - 8
x = 4 - 8
x = -4
Therefore, when x = -4, the square root of 8+x becomes rational because it can be expressed as sqrt(8+(-4)) = 0.
b) To make the number irrational, the square root of 8+x cannot be expressed as a fraction. An irrational number is any number that cannot be expressed as the ratio of two integers.
For the square root of 8+x to be irrational, x needs to be such that it cannot be simplified or expressed as a fraction.
By choosing a value for x that cannot be written in a simplified fraction form, we can make the square root of 8+x irrational. One such example would be x = 1.
So, for x = 1, the square root of 8+x is an irrational number because sqrt(8+1) = sqrt(9) = 3, which cannot be expressed as a fraction.
a) To make the square root of 8+x a rational number, we need to make sure that the expression inside the square root is a perfect square. A perfect square is a number that can be obtained by squaring a whole number.
In this case, we need to find a value for x that makes 8+x a perfect square. One way to approach this is to start with a perfect square and subtract 8 from it. Let's consider the perfect square 16.
If we substitute x = 16 - 8 into the expression, we have:
sqrt(8 + 16 - 8) = sqrt(16) = 4
So, when x = 16, the square root of 8+x becomes sqrt(16) = 4, which is a rational number because it can be expressed as a ratio of two integers (4/1, for example).
b) To make the square root of 8+x an irrational number, we need to find a value for x that makes 8+x not a perfect square.
Let's consider a non-perfect square such as x = 7. Substituting this value into the expression, we have:
sqrt(8 + 7) = sqrt(15)
Here, sqrt(15) is an irrational number because it cannot be expressed as a fraction and its decimal representation goes on infinitely without repeating. So, when x = 7, the square root of 8+x becomes an irrational number.