To determine the value of [tex]\( 27^{\frac{2}{3}} \)[/tex], we can follow these steps:
### Step 1: Understand the Exponent
The expression [tex]\( 27^{\frac{2}{3}} \)[/tex] can be interpreted as taking the cube root of 27 and then squaring the result:
[tex]\[
27^{\frac{2}{3}} = (27^{\frac{1}{3}})^2
\][/tex]
### Step 2: Calculate the Cube Root
Firstly, we need to find the cube root of 27. The cube root of a number [tex]\( x \)[/tex] is a number [tex]\( y \)[/tex] such that [tex]\( y^3 = x \)[/tex]. For [tex]\( 27 \)[/tex]:
[tex]\[
27^{\frac{1}{3}} = 3
\][/tex]
This is because [tex]\( 3^3 = 27 \)[/tex].
### Step 3: Square the Result
Next, we take the result from the previous step and square it:
[tex]\[
(27^{\frac{1}{3}})^2 = 3^2 = 9
\][/tex]
### Conclusion
Therefore, the value of [tex]\( 27^{\frac{2}{3}} \)[/tex] is:
[tex]\[
27^{\frac{2}{3}} = 9
\][/tex]
Hence:
[tex]\[
27^{\frac{2}{3}} \approx 8.999999999999998
\][/tex]
This numerical result is accurate, acknowledging the presence of minor floating-point precision in calculations.
