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consider -/8+x
a) Find a value for x that would make this number rational
b) find a value that would make this number irrational
a) To make the number rational, we need the expression to be a terminating or repeating decimal. This means we need to find a value for x such that the expression results in a fraction.
In order to achieve this, we can express the expression as a fraction by finding a common denominator for 8 and x. The common denominator is 8x.
The expression becomes: 8/8x + x/8x
Combining both terms, we get: (8 + x) / 8x
Now, in order for the expression to be rational, 8 + x must be divisible by 8x.
One possible value for x that satisfies this condition is x = 1.
Plugging x = 1 into the expression, we get: (8 + 1)/8(1) = 9/8, which is a rational number.
b) To make the number irrational, we need the expression to result in a decimal that cannot be expressed as a fraction.
One way to achieve this is to choose a value for x that makes the expression equal to an irrational number, such as π or √2.
For example, if we choose x = π, the expression becomes: 8/8π + π/8π
Simplifying, we get: (8π + π) / 8π
This expression cannot be further simplified into a fraction and therefore the resulting number is irrational.
a) To find a value for x that would make the number rational, we need to check if the square root of the number is rational.
The number is given as -8+x. Let's assume that -8+x is a perfect square, so we have (-8+x) = (sqrt(k))^2 for some integer k.
Expanding the equation, we get -8+x = k.
Solving for x, we find x = k + 8.
So, any value of x that can make the number (-8+x) a perfect square (k + 8)^2 will make the number rational.
b) To find a value that would make the number irrational, we need to choose a value for x that will not result in a perfect square when added to -8.
Choosing x = any non-perfect square number will make the number (-8+x) irrational.
For example, if we let x = 2, then the number becomes -8 + 2 = -6, which is irrational.