Final answer:
To simplify ((-3)^2)^3, we raise -3 to the 2nd power and then take that result and raise it to the 3rd power, effectively multiplying the exponents, resulting in (-3)^6 or 729. The correct answer is option C) (-3)^6.
Step-by-step explanation:
The question asks to simplify the expression ((-3)^2)^3. To simplify such expressions, you need to follow the rules of exponents. When you raise a power to another power, you multiply the exponents. In this case, we have (-3) squared, which is (-3)^2, being cubed, which means we multiply the exponent 2 by 3.
Here's the step-by-step process:
- Start with the innermost parentheses, (-3)^2, which means -3 multiplied by itself: -3 × -3 = 9.
- Now take this result, which is 9, and raise it to the power of 3 since the original expression was ((-3)^2)^3. So we have (9)^3.
- 9 cubed is 9 × 9 × 9 = 729. Therefore, the expression simplifies to 729.
However, if we approach this problem using the exponent multiplication rule and treat the expression as a power of a power, we get:
(((-3)^2)^3 = (-3)^(2×3) = (-3)^6).
Since -3 raised to an even power is always positive, the answer is still 729. Hence the correct answer is option C) ((-3)^6).