Try using the rule $a^{bc} = (a^b)^c$ on $(0.77)^{5x}$. What does the base become after you rewrite it this way?

Once you have $(0.77)^{5x}$ written as something like $(\text{base})^x$, set that base equal to $1 - \frac{q}{100}$ and solve for $q$.

You will need a numerical value for $(0.77)^5$. After you find it, compute $100\bigl(1 - (0.77)^5\bigr)$ and then pick the closest answer choice.

In a Desmos expression line, type (0.77)^5 and note the decimal value that Desmos displays; this is the base that must equal $1 - \frac{q}{100}$.

In a new expression line, type 100*(1 - (0.77)^5). The numerical result is the approximate value of $q$; compare this value to the answer choices and select the closest one.

We are given

and we want it in the form

To compare these, we want both expressions to be something raised to the power $x$ (not $5x$). Use the exponent rule $a^{bc} = (a^b)^c$.

Apply $a^{bc} = (a^b)^c$ to $(0.77)^{5x}$:

So we can write

This shows that in the desired form, the base must be $(0.77)^5$.

We now match the base of the rewritten expression to the form in the problem:

Solve for $q$:

So $q$ is $100$ times the difference between $1$ and $(0.77)^5$.

Compute $(0.77)^5$ (using a calculator or careful multiplication):

• $0.77^2 \approx 0.5929$
• $0.77^3 \approx 0.4565$
• $0.77^4 \approx 0.3515$
• $0.77^5 \approx 0.2707$

Then

The answer choice closest to $72.93$ is 73, so $q \approx 73$.