Let's go through this step-by-step to clarify the calculations and correct the notations.
### Understanding [tex]$\sqrt[3]{9}$[/tex] and [tex]$9^{\frac{1}{3}}$[/tex]
The cube root of a number [tex]\( a \)[/tex] can be written as [tex]\( \sqrt[3]{a} \)[/tex] or equivalently as [tex]\( a^{\frac{1}{3}} \)[/tex]. This equivalence comes from the properties of exponents and radicals.
#### Verifying [tex]$\left(9^{\frac{1}{3}}\right)^3$[/tex]
To understand why [tex]\(\left(9^{\frac{1}{3}}\right)^3 = 9\)[/tex], we use properties of exponents.
1. Exponentiation Property:
[tex]\[
\left(a^{b}\right)^c = a^{b \cdot c}
\][/tex]
When we apply this property to our expression:
[tex]\[
\left(9^{\frac{1}{3}}\right)^3 = 9^{\left(\frac{1}{3} \cdot 3\right)}
\][/tex]
2. Simplifying the Exponent:
[tex]\[
\frac{1}{3} \cdot 3 = 1
\][/tex]
Hence:
[tex]\[
9^{\frac{1}{3} \cdot 3} = 9^1
\][/tex]
3. Final Calculation:
[tex]\[
9^1 = 9
\][/tex]
Thus, we've shown step-by-step that:
[tex]\[
\left(9^{\frac{1}{3}}\right)^3 = 9
\][/tex]
### Incorrect Notations from the Question
Let's correct the provided notations:
1. Incorrect Addition of Exponents:
[tex]\[
\left(9^{\frac{1}{3}}\right)^3 \neq 9^{\left(\frac{1}{3} + 3\right)}
\][/tex]
2. Correct Multiplication of Exponents:
[tex]\[
\left(9^{\frac{1}{3}}\right)^3 = 9^{\left(\frac{1}{3} \cdot 3\right)} = 9^1 = 9
\][/tex]
3. Incorrect Subtraction of Exponents:
[tex]\[
\left(9^{\frac{1}{3}}\right)^3 \neq 9^{\left(\frac{1}{3} - 3\right)}
\][/tex]
### Conclusion
After correctly applying the properties of exponents, we have explained and verified why:
[tex]\[
\left(9^{\frac{1}{3}}\right)^3 = 9
\][/tex]
The final mathematical statement that accurately represents this process is:
[tex]\[
\left(9^{\frac{1}{3}}\right)^3 = 9^{\left(\frac{1}{3} \cdot 3\right)} = 9^1 = 9
\][/tex]
