Understanding the Exponential Equation

The problem asks us to first solve an exponential equation to find the value of 'x', and then use that value to calculate another expression involving the same base.

The given exponential equation is:

$ \left(\frac{5}{3}\right)^{x-3} = \frac{3125}{243} $

Our goal is to find the value of $x$ from this equation.

Expressing the Right Side as a Power

To solve for $x$, we need both sides of the equation to have the same base. The base on the left side is $\frac{5}{3}$. We need to express the right side, $\frac{3125}{243}$, as a power of $\frac{5}{3}$.

Let's look at the powers of 5 and 3:

• $5^1 = 5$
• $5^2 = 25$
• $5^3 = 125$
• $5^4 = 625$
• $5^5 = 3125$

• $3^1 = 3$
• $3^2 = 9$
• $3^3 = 27$
• $3^4 = 81$
• $3^5 = 243$

Using these, we can rewrite $\frac{3125}{243}$ as:

$ \frac{3125}{243} = \frac{5^5}{3^5} = \left(\frac{5}{3}\right)^5 $

Solving for the Exponent 'x'

Now we can substitute this back into the original equation:

$ \left(\frac{5}{3}\right)^{x-3} = \left(\frac{5}{3}\right)^5 $

Since the bases are the same ($\frac{5}{3}$), the exponents must be equal:

$ x-3 = 5 $

To find $x$, we add 3 to both sides:

$ x = 5 + 3 $ $ x = 8 $

So, the value of $x$ is 8.

Calculating the Final Value

The question asks for the value of $\left(\frac{5}{3}\right)^{\frac{x}{2}}$. We now know that $x=8$. Let's substitute this value:

$ \left(\frac{5}{3}\right)^{\frac{x}{2}} = \left(\frac{5}{3}\right)^{\frac{8}{2}} $

Simplify the exponent:

$ \frac{8}{2} = 4 $

So, we need to calculate:

$ \left(\frac{5}{3}\right)^4 $

Calculate the value:

$ \left(\frac{5}{3}\right)^4 = \frac{5^4}{3^4} = \frac{625}{81} $

Therefore, the value of $\left(\frac{5}{3}\right)^{\frac{x}{2}}$ is $\frac{625}{81}$.