Question
Which equation justifies why nine to the one third power equals the cube root of nine?
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
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
Answers
GPT-4o mini
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.
**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.