Target Test Prep EA Quant Challenge Problem 5

Solution:

We should recognize that we have subtraction of bases with exponents. This means before we can combine the equation's terms, we need to factor out common factors. However, to help see what we can factor out, we can rewrite the equation.

$\begin{array}{l}\Rightarrow 5^{\mathrm{x}}-5^{\mathrm{x}-1}=500\\ \Rightarrow 5^{\mathrm{x}}-{\left(5\right)}^{\mathrm{x}}{\left(5\right)}^{-1}=500\end{array}$

Now we can easily see to factor out the common term of 5^x. We now have:

$\begin{array}{l}\Rightarrow 5^{\mathrm{x}}-{\left(5\right)}^{\mathrm{x}}{\left(5\right)}^{-1}=500\\ \Rightarrow 5^{\mathrm{x}}\left(1-5^{-1}\right)=500\\ \Rightarrow 5^{\mathrm{x}}\left(1-\frac{1}{5}\right)=500\\ \Rightarrow 5^{\mathrm{x}}\times \frac{4}{5}=500\end{array}$

Our next step is to break down all the values into prime factors. This will make canceling out much easier.

$\begin{array}{l}\Rightarrow 5^{\mathrm{x}}\times \frac{4}{5}=500\to 5^{\mathrm{x}}\times \frac{2^2}{5}=5^3\times 2^2\\ \Rightarrow 5^{\mathrm{x}}=\frac{5^3\times 2^2\times 5}{2^2}\\ \Rightarrow 5^{\mathrm{x}}=5^4\to \mathrm{x}=4\end{array}$

Finally, we have to solve for (x – 1)², so (4 – 1)² = 3² = 9