Question

select all the expressions that are equivalent to $\frac{9^{3}}{3^{3}}$.$3^{3}$$\frac{1}{3^{3}}$$(3^{3})^{1}$$\frac{3^{3}}{3}$

Explanation:

Step1: Simplify the original expression

First, rewrite 9 as $3^2$:
$$\frac{9^3}{3^3} = \frac{(3^2)^3}{3^3}$$
Use exponent rule $(a^m)^n = a^{m \times n}$:
$$\frac{(3^2)^3}{3^3} = \frac{3^{6}}{3^3}$$
Use exponent rule $\frac{a^m}{a^n} = a^{m-n}$:
$$\frac{3^{6}}{3^3} = 3^{6-3} = 3^3 = 27$$

Step2: Evaluate each option

Option1: $3^3$

$3^3 = 27$, matches the original value.

Option2: $\frac{1}{3^3}$

$\frac{1}{3^3} = \frac{1}{27}$, does not match.

Option3: $(3^3)^1$

Use exponent rule $(a^m)^n = a^{m \times n}$:
$$(3^3)^1 = 3^{3 \times 1} = 3^3 = 27$$, matches the original value.

Option4: $\frac{3^3}{3}$

Use exponent rule $\frac{a^m}{a^n} = a^{m-n}$:
$$\frac{3^3}{3} = 3^{3-1} = 3^2 = 9$$, does not match.

Answer:

A. $3^3$, C. $(3^3)^1$