Certainly! Let's start with the given equation and work through it step by step to convert it into a form without any rational exponents.
Given equation:
[tex]\[ T = A^{3/2} \][/tex]
First, let's understand what the exponent [tex]\( \frac{3}{2} \)[/tex] means. It signifies that [tex]\( A \)[/tex] is raised to the power of 3 and then the square root is taken (or vice versa, since the operations will yield the same result):
[tex]\[ T = (A^{3})^{1/2} \][/tex]
However, we need an alternative approach to eliminate the rational exponent.
Here’s a step-by-step strategy:
1. Raise both sides of the equation to a power that will eliminate the fraction:
[tex]\[ \left(T\right)^{2/3} = \left(A^{3/2}\right)^{2/3} \][/tex]
2. Simplify the right-hand side:
[tex]\[ \left(T\right)^{2/3} = A^{(3/2) \cdot (2/3)} \][/tex]
3. Calculate the exponents:
[tex]\[ \left(\frac{3}{2} \cdot \frac{2}{3} = 1\right) \][/tex]
So, the right-hand side simplifies to just [tex]\( A \)[/tex]:
[tex]\[ \left(T\right)^{2/3} = A \][/tex]
Thus, we have successfully converted the original equation into a form without any rational exponents:
[tex]\[ \boxed{T^{2/3} = A} \][/tex]
This rewritten equation expresses [tex]\( T \)[/tex] and [tex]\( A \)[/tex] without involving any rational exponents, making it much simpler to handle in further mathematical contexts.
