Question

Identify all equivalent expressions:

(12a^3)^0(14b)^2


Responses

(1/2)^0×a^3x0×(14)^2×b^2

1×1×1/4×1/4×b×b

0


(1/2)^0×a^3+0×(1/4)^2×b^2

1/16b^2

0×0×1/4×1/4×b×b

Answer 1

To simplify the expression \((12a^3)^0(14b)^2\), we can break it down step by step.

1. **Evaluate \((12a^3)^0\)**:
Any non-zero expression raised to the power of 0 equals 1.
Thus, \((12a^3)^0 = 1\).

2. **Evaluate \((14b)^2\)**:
This involves squaring both the coefficient and the variable:
\((14b)^2 = (14^2)(b^2) = 196b^2\).

3. **Combine the results**:
Finally, since \((12a^3)^0 = 1\), the entire expression simplifies to:
\[
1 \cdot 196b^2 = 196b^2.
\]

Now, let's evaluate the provided responses and see which can be simplified to \(196b^2\):

1. **Response 1**: \((1/2)^0 \times a^{3 \cdot 0} \times (14)^2 \times b^2\)
- This simplifies to \(1 \cdot 1 \cdot 196 \cdot b^2 = 196b^2\).

2. **Response 2**: \(1 \times 1 \times 1/4 \times 1/4 \times b \times b\)
- This simplifies to \(1 \times 1 \times \frac{1}{16} \times b^2 = \frac{1}{16}b^2\), which is not \(196b^2\).

3. **Response 3**: \(0\)
- This is not equivalent to \(196b^2\).

4. **Response 4**: \((1/2)^0 \times a^{3 + 0} \times (1/4)^2 \times b^2\)
- This simplifies to \(1 \cdot a^3 \cdot \frac{1}{16} \cdot b^2 = \frac{1}{16}a^3b^2\), which is not equal to \(196b^2\).

5. **Response 5**: \(1/16b^2\)
- This is not equal to \(196b^2\).

6. **Response 6**: \(0 \times 0 \times \frac{1}{4} \times \frac{1}{4} \times b \times b\)
- This is equal to \(0\), not \(196b^2\).

### Conclusion

The only equivalent expression to the original expression \((12a^3)^0(14b)^2\) is:

- **Response 1**: \((1/2)^0 \times a^{3 \cdot 0} \times (14)^2 \times b^2\)

Thus, the answer is:

**Equivalent expression**:
\((1/2)^0 \times a^{3 \cdot 0} \times (14)^2 \times b^2\)