If $(\frac{5}{3})^{x-3} = \frac{3125}{243}$, find the value of $(\frac{5}{3})^{\frac{x}{2}}$.

Question

Prepp Team · Accepted Answer

Solving Exponential Equation

We are given the equation $(\frac{5}{3})^{x-3} = \frac{3125}{243}$ and asked to find the value of $(\frac{5}{3})^{\frac{x}{2}}$.

First, let's express the number $\frac{3125}{243}$ as a power of the base $\frac{5}{3}$.

We find the prime factors of the numerator and the denominator:

Therefore, we can write $\frac{3125}{243}$ as:

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

Our original equation now becomes:

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

Since the bases on both sides of the equation are the same ($\frac{5}{3}$), the exponents must be equal.

So, we set the exponents equal to each other:

$ x - 3 = 5 $

To find the value of $x$, we add 3 to both sides of the equation:

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

Now we know that $x = 8$.

The question asks for the value of $(\frac{5}{3})^{\frac{x}{2}}$. We substitute the value of $x$ we found:

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

Simplify the exponent:

$ \left(\frac{5}{3}\right)^{\frac{8}{2}} = \left(\frac{5}{3}\right)^4 $

Now, we calculate the value of $(\frac{5}{3})^4$:

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

So, the value is:

$ \frac{625}{81} $

The value of $(\frac{5}{3})^{\frac{x}{2}}$ is $\frac{625}{81}$.

Prepp Team · Answer

$\frac{5}{3}$

Prepp Team · Answer

$\frac{125}{27}$

Prepp Team · Answer

$\frac{25}{9}$