To determine which statements are true regarding the solutions to the given equations, let's analyze them one by one.
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For the equation \( x^2 = 2564 \):
- The solutions are \( x = \sqrt{2564} \) and \( x = -\sqrt{2564} \).
- Since \( \sqrt{2564} \approx 50.64 \), we can conclude that the approximate solutions are \( x \approx 50.64 \) and \( x \approx -50.64 \).
- If we actually compute, the square roots give us \( x \approx 50.64 \) and negative of that.
The given options state that the solutions are \( x = -58 \) and \( x = 58 \). Therefore, this statement is false.
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The statement that the only solution to the equation \( x^2 = 2564 \) is \( x = 58 \) is also false since \( \pm \sqrt{2564} \) are both solutions.
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For the equation \( x^3 = -2764 \):
- This can be solved by taking the cube root: \( x = \sqrt[3]{-2764} \).
- The cube root of a negative number is negative. Calculating \( \sqrt[3]{-2764} \) gives \( x \approx -34 \).
So the statement, "The only solution to the equation \( x^3 = -2764 \) is \( x = -34 \)" is true.
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The equation \( x^3 = -27/64 \) implies that \( x^3 \) has one real solution, \( x = -3/4 \) (since \( (-3/4)^3 = -27/64 \)). Thus it follows that the statement "The only solution to the equation \( x^3 = -27/64 \) is \( x = -3/4 \)" is also true.
So, the two statements that are true are:
- "The only solution to the equation \( x^3 = -2764 \) is \( x = -34 \)"
- "The only solution to the equation \( x^3 = -27/64 \) is \( x = -3/4 \)"