To understand why [tex]\( \sqrt[3]{9} \)[/tex] is equal to [tex]\( 9^{\frac{1}{3}} \)[/tex], let's delve into the properties of exponents and radicals.
By definition, the cube root of a number is a value that, when raised to the power of 3, gives the original number. Mathematically, this can be expressed as:
[tex]\[ \sqrt[3]{a} = a^{\frac{1}{3}} \][/tex]
In our case, [tex]\( a = 9 \)[/tex]. Thus,
[tex]\[ \sqrt[3]{9} = 9^{\frac{1}{3}} \][/tex]
Now, to verify that [tex]\( 9^{\frac{1}{3}} \)[/tex] indeed represents the cube root of 9, we can raise it to the power of 3 and check if it equals 9. Let's perform this step-by-step:
1. Start with [tex]\( 9^{\frac{1}{3}} \)[/tex].
2. Raise this expression to the power of 3:
[tex]\[ \left(9^{\frac{1}{3}}\right)^3 \][/tex]
3. When you raise a power to another power, you multiply the exponents. Therefore:
[tex]\[ \left(9^{\frac{1}{3}}\right)^3 = 9^{\left(\frac{1}{3} \times 3\right)} \][/tex]
4. Simplify the exponent:
[tex]\[ \frac{1}{3} \times 3 = 1 \][/tex]
5. Hence:
[tex]\[ 9^{\left(\frac{1}{3} \times 3\right)} = 9^1 \][/tex]
6. Finally:
[tex]\[ 9^1 = 9 \][/tex]
Thus, we have shown that:
[tex]\[ \left(9^{\frac{1}{3}}\right)^3 = 9 \][/tex]
This confirms that [tex]\( 9^{\frac{1}{3}} \)[/tex] is indeed the cube root of 9, and therefore [tex]\(\sqrt[3]{9} = 9^{\frac{1}{3}} \)[/tex].
The choice in your question showing the correct mathematical steps is:
[tex]\[ \left(9^{\frac{1}{3}}\right)^3=9^{\left(\frac{1}{3}\times 3\right)}=9^1=9 \][/tex]
