To determine which of the given equations are equivalent to [tex]\( T = A^{1.5} \)[/tex], we need to understand how different forms can represent the same expression. Let’s analyze each of the given equations step-by-step.
### Original Equation
The given base equation is:
[tex]\[ T = A^{1.5} \][/tex]
Since [tex]\( 1.5 \)[/tex] is the same as [tex]\( \frac{3}{2} \)[/tex], we can rewrite it as:
[tex]\[ T = A^{\frac{3}{2}} \][/tex]
### Equation 1:
[tex]\[ T = A^{3/2} \][/tex]
This is identical to the original equation written as [tex]\( A^{1.5} \)[/tex], since [tex]\( 1.5 = \frac{3}{2} \)[/tex]. Therefore, this equation is equivalent to the original equation.
### Equation 2:
[tex]\[ T = \left(A^{1/2}\right)^3 \][/tex]
We can use the property of exponents, [tex]\( (A^{m})^n = A^{m \cdot n} \)[/tex], to simplify this:
[tex]\[ \left(A^{1/2}\right)^3 = A^{(1/2) \cdot 3} = A^{3/2} \][/tex]
This matches the original equation [tex]\( T = A^{1.5} \)[/tex]. Therefore, this equation is also equivalent.
### Equation 3:
[tex]\[ T = (\sqrt{A})^3 \][/tex]
Since [tex]\( \sqrt{A} \)[/tex] is the same as [tex]\( A^{1/2} \)[/tex], we can rewrite the equation as:
[tex]\[ T = (A^{1/2})^3 \][/tex]
Following the same steps as in Equation 2:
[tex]\[ (A^{1/2})^3 = A^{(1/2) \cdot 3} = A^{3/2} \][/tex]
This is equivalent to the original equation [tex]\( T = A^{1.5} \)[/tex]. Therefore, this equation is equivalent as well.
### Equation 4:
[tex]\[ T = \sqrt{A^3} \][/tex]
We can use the property of square roots, which is [tex]\( \sqrt{B} = B^{1/2} \)[/tex], to simplify this:
[tex]\[ \sqrt{A^3} = (A^3)^{1/2} \][/tex]
Using the properties of exponents [tex]\( (A^m)^n = A^{m \cdot n} \)[/tex]:
[tex]\[ (A^3)^{1/2} = A^{3 \cdot (1/2)} = A^{3/2} \][/tex]
This matches the original equation [tex]\( T = A^{1.5} \)[/tex]. Therefore, this equation is also equivalent.
### Conclusion
All the provided equations are equivalent to the original equation [tex]\( T = A^{1.5} \)[/tex]:
- [tex]\( T = A^{3/2} \)[/tex]
- [tex]\( T = \left(A^{1/2}\right)^3 \)[/tex]
- [tex]\( T = (\sqrt{A})^3 \)[/tex]
- [tex]\( T = \sqrt{A^3} \)[/tex]
Therefore, all the given equations (1, 2, 3, and 4) are equivalent to the original equation [tex]\( T = A^{1.5} \)[/tex]. The correct answer is:
[tex]\[ \boxed{[1, 2, 3, 4]} \][/tex]
