Is xy Positive?: Analyzing Mathematical Statements

The objective is to determine whether the product \(xy\) is positive. We must evaluate if the given statements, individually or collectively, provide enough information to answer this question.

Statement I Analysis: Value of x

Statement I provides the definition for \(x\):

\(x = \sqrt[3]{-0.19683}\)

To find the value of \(x\), we need to compute the cube root of \(-0.19683\). Since the cube root of a negative number is itself negative, \(x\) will be negative. We observe that \(27^3 = 19683\). Therefore, \(\sqrt[3]{0.19683} = 0.57\). Consequently, the value of \(x\) is:

\(x = -0.57\)

Statement I successfully determines \(x\) as a specific negative number. However, it offers no information about the value or sign of \(y\). Without knowing \(y\), we cannot ascertain the sign of the product \(xy\).

Conclusion: Statement I alone is insufficient to answer the question.

Statement II Analysis: Relationship for y

Statement II defines \(y\) in relation to \(x\):

\(y = \sqrt[3]{x}\)

This statement establishes a mathematical relationship between \(y\) and \(x\). It tells us that \(y\) is the cube root of \(x\). However, this statement does not provide a specific numerical value for \(x\) or \(y\). We lack the necessary information to determine the sign of \(x\), and consequently, the sign of \(y\) or their product \(xy\).

Conclusion: Statement II alone is insufficient to answer the question.

Combined Statements Analysis: Evaluating xy

By considering both statements together, we can proceed as follows:

• From Statement I, we have determined that \(x = -0.57\).
• From Statement II, we know that \(y = \sqrt[3]{x}\).

Substituting the value of \(x\) from Statement I into the expression for \(y\) given in Statement II:

\(y = \sqrt[3]{-0.57}\)

The cube root of a negative number is always negative. Thus, \(y\) is a negative value.

We now have the signs of both \(x\) and \(y\):

• \(x\) is negative (\(-0.57\)).
• \(y\) is negative (\(\sqrt[3]{-0.57}\)).

To determine if \(xy\) is positive, we consider the product of two negative numbers:

\(xy = (\text{negative value}) \times (\text{negative value})\)

The product of two negative numbers is always positive.

Conclusion: Using both statements together, we can definitively conclude that \(xy\) is positive.

Final Sufficiency Determination

Since neither Statement I nor Statement II, when considered individually, is sufficient to answer the question "Is \(xy\) positive?", but combining both statements allows for a clear determination, the correct approach is to use both statements together.