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.
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
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