Question
find domain of (x-5)^(1/3)/(x^3-5)
Answers
Answered by
GPT 3.5
To find the domain of the given expression, we need to identify any values of x that would result in division by zero or taking an even root of a negative number.
First, let's consider the denominator, x^3 - 5. The expression would be undefined if x^3 - 5 = 0. Solving for x:
x^3 = 5
x = ∛5
Next, let's consider the numerator, (x - 5)^(1/3). Taking the cube root of a negative number would result in a complex number, so we need to make sure x - 5 is not negative.
x - 5 ≥ 0
x ≥ 5
Therefore, the domain of the expression is all real numbers greater than or equal to 5, or [5, ∞).
First, let's consider the denominator, x^3 - 5. The expression would be undefined if x^3 - 5 = 0. Solving for x:
x^3 = 5
x = ∛5
Next, let's consider the numerator, (x - 5)^(1/3). Taking the cube root of a negative number would result in a complex number, so we need to make sure x - 5 is not negative.
x - 5 ≥ 0
x ≥ 5
Therefore, the domain of the expression is all real numbers greater than or equal to 5, or [5, ∞).