Sure, let's work through this step-by-step to determine which equations are equivalent to [tex]\( T = A^{1.5} \)[/tex]. Remember that [tex]\( 1.5 \)[/tex] is the same as [tex]\( \frac{3}{2} \)[/tex].
1. Equation: [tex]\( T = A^{3 / 2} \)[/tex]
Simplifying this:
[tex]\[
T = A^{3 / 2}
\][/tex]
Since [tex]\( A^{1.5} = A^{3 / 2} \)[/tex], this equation is equivalent to [tex]\( T = A^{1.5} \)[/tex].
2. Equation: [tex]\( T = \left(A^{1 / 2}\right)^3 \)[/tex]
Simplifying this:
[tex]\[
T = \left(A^{1 / 2}\right)^3 = A^{(1 / 2) \cdot 3} = A^{3 / 2}
\][/tex]
Again, since [tex]\( A^{1.5} = A^{3 / 2} \)[/tex], this equation is equivalent to [tex]\( T = A^{1.5} \)[/tex].
3. Equation: [tex]\( T = (\sqrt{A})^3 \)[/tex]
Simplifying this:
[tex]\[
T = (\sqrt{A})^3 = (A^{1 / 2})^3 = A^{(1 / 2) \cdot 3} = A^{3 / 2}
\][/tex]
Here, we have [tex]\( A^{3/2} \)[/tex] which is equivalent to [tex]\( A^{1.5} \)[/tex]. Therefore, this equation is equivalent to [tex]\( T = A^{1.5} \)[/tex].
4. Equation: [tex]\( T = \sqrt{A^3} \)[/tex]
Simplifying this:
[tex]\[
T = \sqrt{A^3} = (A^3)^{1 / 2} = A^{3 \cdot (1 / 2)} = A^{3 / 2}
\][/tex]
This canonicalizes to [tex]\( A^{1.5} = A^{3 / 2} \)[/tex]. So, this equation is also equivalent to [tex]\( T = A^{1.5} \)[/tex].
To summarize based on our step-by-step simplifications:
All the equations given here are equivalent to [tex]\( T = A^{1.5} \)[/tex].
The simplified results indicate that all of the given equations are indeed equivalent to [tex]\( T = A^{1.5} \)[/tex], i.e., [tex]\( [1, 1, 1, 1] \)[/tex].
