Question

Which of the following equations can be rewritten to have common bases on both sides? Select all that apply.(1 point) Responses 3x=4x+1 3 to the x th power is equal to 4 raised to the x plus 1 power 3x=92x−1 3 to the x th power is equal to 9 raised to the 2 x minus 1 power 43x−1=64x 4 raised to the 3 x minus 1 power is equal to 64 to the x th power 2x=6 2 to the x th power is equal to 6 81=9x

Answer 1

To determine which equations can be rewritten to have common bases on both sides, we should look for the potential to express the numbers as powers of the same base.

1. **3^x = 4^(x + 1)**:
- This equation cannot be rewritten to have the same base, as 3 and 4 are different bases with no common base representation.

2. **3^x = 9^(2x - 1)**:
- Here, 9 can be rewritten as \(3^2\). So, \(9^{(2x - 1)} = (3^2)^{(2x - 1)} = 3^{2(2x - 1)} = 3^{4x - 2}\). Therefore, the equation can be rewritten as \(3^x = 3^{4x - 2}\), which has a common base.

3. **4^(3x - 1) = 64^x**:
- The number 64 can be rewritten as \(4^3\). So, \(64^x = (4^3)^x = 4^{3x}\). Thus, it can be rewritten as \(4^{(3x - 1)} = 4^{3x}\), which has a common base.

4. **2^x = 6**:
- This equation cannot be rewritten to have a common base, as 6 cannot be expressed as a power of 2.

5. **81 = 9^x**:
- Here, 81 can be rewritten as \(9^2\). So, the equation can be written as \(9^2 = 9^x\), which has a common base.

Thus, the equations that can be rewritten to have common bases are **3^x = 9^(2x - 1)**, **4^(3x - 1) = 64^x**, and **81 = 9^x**.