Question

The expressions \(3 x^{2}\) and \((3 x)^{2}\) do not have the same meaning. Explain the difference.

Short answer

The expression \(3x^{2}\) results in \(3x^2\), squaring only the \(x\), whereas \((3x)^{2}\) applies the square to the entire term within the parentheses, resulting in \(9x^2\). Thus, the placement of the exponent and parentheses does matter and changes the meaning of the expressions.

Step-by-step solution

Analyzing the first expression

In the expression \(3 x^{2}\), the exponent 2 only applies to the variable \(x\), because no parentheses are altering the order of operations. Therefore, we leave the coefficient 3 as it is and only square \(x\). The expression \(3 x^{2}\) stands for \(3 \cdot x \cdot x\) or \(3x^2\).

Analyzing the second expression

In the expression \((3 x)^{2}\), the parentheses indicate that the squaring applies to the entire term within – that includes both the 3 and the \(x\). According to mathematical order of operations, parentheses come first, then exponentiation. Therefore, the expression \((3 x)^{2}\) should be interpreted as \((3x)\cdot (3x)\) or \(9x^2\).

Conclusion and comparison

Ultimately, while \(3 x^{2}\) results in \(3x^2\), \((3x)^{2}\) results in \(9x^2\). Therefore, despite their similar appearance, these two expressions do not mean the same thing because the order of operations interprets them differently.