Let's break down the given problem into smaller, more digestible steps while ensuring each part is understood.
Firstly, we need to find [tex]\(\sqrt[3]{27^2}\)[/tex]. Now, to make it easier, let's consider the properties of exponents and cube roots.
1. Step 1: Calculation of [tex]\(27\)[/tex] raised to the power [tex]\(2\)[/tex]
We start by calculating the immediate exponentiation:
[tex]\[
27^2 = 27 \times 27 = 729
\][/tex]
2. Step 2: Calculation of the cube root of [tex]\(729\)[/tex]
We are tasked with finding the cube root of [tex]\(729\)[/tex]:
[tex]\[
\sqrt[3]{729}
\][/tex]
Now, instead of directly calculating [tex]\(\sqrt[3]{729}\)[/tex], let's approach the problem using the property of exponents:
[tex]\[
\sqrt[3]{27^2} = \left(\sqrt[3]{27}\right)^2
\][/tex]
3. Step 3: Calculation of the cube root of [tex]\(27\)[/tex]
The cube root of [tex]\(27\)[/tex] is:
[tex]\[
\sqrt[3]{27} = 3
\][/tex]
4. Step 4: Squaring the cube root result
Next, square the result obtained from the cube root:
[tex]\[
(3)^2 = 3 \times 3 = 9
\][/tex]
Therefore, the whole expression [tex]\(\sqrt[3]{27^2}\)[/tex] simplifies to:
[tex]\[
\sqrt[3]{27^2} = (\sqrt[3]{27})^2 = 3^2 = 9
\][/tex]
So, the final result is:
[tex]\[
\boxed{9}
\][/tex]
