We are given the expressions shown below and asked to find the one that is different. We will label each of them in order to tell them apart.

A) (327)^{2} C) 3^{2} B) 2 7^{\frac{2}{3}} D) (227)^{3}

Since these expressions involve rational exponents and radicals, it will be useful to review how can we relate them.

Equivalence Beween Rational Exponents and Radicals
Let $a$ be a real number, and $n$ and $m$ are two positive integers. We can express a power with a rational exponent as a radical, as indicated below. $a^{\frac{m}{n}} = (n a)^{m}$

According to this property, we can rewrite expression

B

as expression

A .

2 7^{\frac{2}{3}} = (327)^{2}

This allows us to evaluate

B, 2 7^{\frac{2}{3}},

by simplifying

A, (327)^{2} .

We can start by simplifying the expression inside the parentheses by using the definition of an

n th

(327)^{2}

Rewrite $27$ as $3^{3}$

(3 3^{3})^{2}

$n a^{n} = a$

3^{2}

Calculate power

9

As we can see, expressions A and B are also equivalent to expression C — they all equal

3^{2} .

Therefore, the expression that is different must be D,

(227)^{3} .

This expression cannot be simplified any more, but we can approximate it using a calculator.

(227)^{3}

Use a calculator

140.296115 \dots

Round to $3$ decimal place(s)

140.296

We just found that D,

(227)^{3} \approx 140.296 .

As we can see, it is clearly different from the other expressions. Therefore, this is the odd one out.

Exercise