Question

1. mrs. tarpley wrote the expression ((27x)^{\frac{1}{3}}) on the board. three students made statements about the expression. which student made a correct statement? dorie: the expression is equivalent to (\frac{1}{(27x)^3}). alonso: the expression is equivalent to (9(sqrt3{x})). nick: the expression is equivalent to the cube root of (27x).

Explanation:

Step1: Recall exponent rules

We know that \(a^{-n}=\frac{1}{a^{n}}\) and \((ab)^{n}=a^{n}b^{n}\), also \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) or \((\sqrt[n]{a})^{m}\). Given the expression \((27x)^{\frac{1}{3}}\).

Step2: Apply the power - of - a - product rule

Using \((ab)^{n}=a^{n}b^{n}\), we can rewrite \((27x)^{\frac{1}{3}}\) as \(27^{\frac{1}{3}}\times x^{\frac{1}{3}}\).

Step3: Simplify \(27^{\frac{1}{3}}\)

We know that \(3^{3}=27\), so \(27^{\frac{1}{3}}=\sqrt[3]{27} = 3\) (since if \(y = a^{\frac{1}{n}}\), then \(y^{n}=a\), and \(3^{3}=27\)). And \(x^{\frac{1}{3}}=\sqrt[3]{x}\) or \((\sqrt[3]{x})^{1}\).

Step4: Rewrite the expression

So \(27^{\frac{1}{3}}\times x^{\frac{1}{3}}=3\times\sqrt[3]{x}=3\sqrt[3]{x}\) or \((3\sqrt[3]{x})\) or we can also write it as \((\sqrt[3]{27x})\) (since \(27^{\frac{1}{3}}x^{\frac{1}{3}}=(27x)^{\frac{1}{3}}=\sqrt[3]{27x}\)) or as \(3x^{\frac{1}{3}}\).

Now let's analyze each student's statement:

• Dorie: Dorie says the expression is equivalent to \(\frac{1}{(27x)^{3}}\). But \(\frac{1}{(27x)^{3}}=(27x)^{- 3}

eq(27x)^{\frac{1}{3}}\), so Dorie is wrong.

• Alonso: Alonso says the expression is equivalent to \(3(\sqrt[3]{x})\). Since \(27^{\frac{1}{3}} = 3\) and \(x^{\frac{1}{3}}=\sqrt[3]{x}\), \((27x)^{\frac{1}{3}}=27^{\frac{1}{3}}x^{\frac{1}{3}} = 3\sqrt[3]{x}\), so Alonso is correct.
• Nick: Nick's statement is incomplete, but from what we can see, if we assume he says it's the cube root of \(27x\), \(\sqrt[3]{27x}=(27x)^{\frac{1}{3}}\), which is also correct. But among the given (from the visible part), Alonso's statement \(3(\sqrt[3]{x})\) is a more simplified and clear correct form.

Answer:

Alonso made a correct statement (the expression \((27x)^{\frac{1}{3}}\) is equivalent to \(3(\sqrt[3]{x})\))