To evaluate [tex]\( 216^{2/3} \)[/tex], we will go through a step-by-step process:
1. Understand the expression: [tex]\( 216^{2/3} \)[/tex] is requesting the number that, when raised to the power of [tex]\( \frac{2}{3} \)[/tex], gives 216.
2. Rewrite the exponent: The exponent [tex]\( \frac{2}{3} \)[/tex] can be interpreted in two steps:
- First, raise 216 to the power of [tex]\( \frac{1}{3} \)[/tex]. This is equivalent to finding the cube root of 216.
- Then, square the result from the previous step.
3. Find the cube root of 216:
- The cube root of 216 is the value which, when cubed, equals 216.
- [tex]\( 6^3 = 6 \times 6 \times 6 = 216 \)[/tex]
- Therefore, [tex]\( \sqrt[3]{216} = 6 \)[/tex].
4. Square the cube root of 216:
- Now we need to square 6, which means raising 6 to the power of 2.
- [tex]\( 6^2 = 6 \times 6 = 36 \)[/tex].
Thus, the value of [tex]\( 216^{2/3} \)[/tex] is:
[tex]\[ 216^{2/3} = 36. \][/tex]
The detailed evaluation confirms the solution to [tex]\( 216^{2/3} \)[/tex] is approximately 36.
