Why is [tex]\sqrt[3]{9}[/tex] equal to [tex]9^{\frac{1}{3}}[/tex]?

A. [tex]\left(9^{\frac{1}{3}}\right)^3=9^{\left(\frac{1}{3}+3\right)}=9^1=9[/tex]

B. [tex]\left(9^{\frac{1}{3}}\right)^3=9^{\left(\frac{1}{3}\right)(3)}=9^1=9[/tex]

C. [tex]\left(9^{\frac{1}{3}}\right)^3=9^{\left(\frac{1}{3}-3\right)}=9^1=9[/tex]

D. [tex]\left(9^{\frac{1}{3}}\right)^3=9^{\left(\frac{1}{3}+3\right)}=9^1=9[/tex]



Answer :

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]