To justify why [tex]\( 9^{\frac{1}{3}} = \sqrt[3]{9} \)[/tex], let's go through the step-by-step process.
Step 1: Understand what [tex]\( 9^{\frac{1}{3}} \)[/tex] and [tex]\( \sqrt[3]{9} \)[/tex] represent.
- [tex]\( 9^{\frac{1}{3}} \)[/tex] represents the cube root of 9.
- [tex]\( \sqrt[3]{9} \)[/tex] is another notation for the cube root of 9.
Step 2: Let’s verify that both expressions are indeed equal.
- When we say [tex]\( 9^{\frac{1}{3}} \)[/tex], we are looking for a number which, when raised to the power of 3, gives us 9.
Step 3: To justify the equivalence, we raise both sides of the equation to the power of 3.
- Start with [tex]\( 9^{\frac{1}{3}} \)[/tex]:
[tex]\[ \left(9^{\frac{1}{3}}\right)^3 \][/tex]
Step 4: Use the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[ \left(9^{\frac{1}{3}}\right)^3 = 9^{\left(\frac{1}{3} \cdot 3\right)} \][/tex]
Step 5: Simplify the exponent [tex]\(\frac{1}{3} \cdot 3\)[/tex]:
[tex]\[ 9^{\left(\frac{1}{3} \cdot 3\right)} = 9^1 = 9 \][/tex]
So, the correct equation that justifies why [tex]\( 9^{\frac{1}{3}} = \sqrt[3]{9} \)[/tex] is:
[tex]\[ \left(9^{\frac{1}{3}}\right)^3=9^{\left(\frac{1}{3} \cdot 3\right)}=9 \][/tex]
Thus, the correct answer is:
[tex]\[ \left(9^{\frac{1}{3}}\right)^3=9^{\left(\frac{1}{3} \cdot 3\right)}=9 \][/tex]
