To determine if [tex]\( w = 3 \)[/tex] is a valid solution for the given equations, we should solve each equation separately.
### Equation A: [tex]\( w^2 = 9 \)[/tex]
1. Recall that squaring a number means multiplying it by itself.
2. [tex]\( w^2 = 9 \)[/tex] implies we need to find a number [tex]\( w \)[/tex] such that when it is squared, the result is 9.
3. The two-square roots of 9 are [tex]\( 3 \)[/tex] and [tex]\( -3 \)[/tex] because:
[tex]\[
3^2 = 9 \quad \text{and} \quad (-3)^2 = 9
\][/tex]
4. Therefore, [tex]\( w = 3 \)[/tex] is indeed one of the solutions to the equation [tex]\( w^2 = 9 \)[/tex].
### Equation B: [tex]\( w^3 = 27 \)[/tex]
1. Here, cubing a number means multiplying the number by itself three times.
2. [tex]\( w^3 = 27 \)[/tex] implies we need to find a number [tex]\( w \)[/tex] such that when it is cubed, the result is 27.
3. The cube root of 27 is [tex]\( 3 \)[/tex] because:
[tex]\[
3^3 = 3 \times 3 \times 3 = 27
\][/tex]
4. Therefore, [tex]\( w = 3 \)[/tex] is the unique real solution to the equation [tex]\( w^3 = 27 \)[/tex].
### Conclusion
Both equations [tex]\( w^2 = 9 \)[/tex] and [tex]\( w^3 = 27 \)[/tex] have [tex]\( w = 3 \)[/tex] as a possible value. Therefore, both A and B are correct answers.
So the final answer is:
A and B
