Which equation justifies why nine to the one third power equals the cube root of nine?

Question

a nine to the one third power all raised to the third power equals nine raised to the one third times three power equals nine
b nine to the one third power all raised to the third power equals nine raised to the one third plus three power equals nine
c nine to the one third power all raised to the third power equals nine raised to the one third minus three power equals nine
d nine to the one third power all raised to the third power equals nine raised to the three minus one third power equals nine

Answer 1

The correct justification for why \( 9^{\frac{1}{3}} \) equals the cube root of 9 can be found in option (a):

a \( (9^{\frac{1}{3}})^3 = 9^{\frac{1}{3} \times 3} = 9 \)

Here’s the breakdown:

1. \( 9^{\frac{1}{3}} \) represents the cube root of 9.
2. Raising \( 9^{\frac{1}{3}} \) to the third power, we multiply the exponents: \( \frac{1}{3} \times 3 = 1 \).
3. Thus, \( (9^{\frac{1}{3}})^3 = 9^1 = 9 \).

This validates that \( 9^{\frac{1}{3}} \) indeed equals the cube root of 9.