Using the distributive property to factor $(5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x)$

Question

I can't seem to understand the distributive property.

Take this:

$$ 5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x$$

becoming this:

$$ 5^x\left(\frac 15 - 1 - 5 +25\right) $$

Help? :D

Answer 1

Every term has a factor of $5^x$ which is factored out.

Also, note that:

$$5^{-1} = \frac 1{5^1} = \frac 15,\quad 5^2 = 25$$

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$$\begin{align} 5^{-1}\cdot \color{blue}{5^x} - (1)\cdot \color{blue}{5^x} - 5\cdot \color{blue}{5^x} + 5^2\cdot \color{blue}{5^x} &= \frac 15\cdot \color{blue}{5^x} -(1)\cdot \color{blue}{5^x} - 5\cdot \color{blue}{5^x}+ 25\cdot \color{blue}{5^x}\\ \\ & =\color{blue}{5^x}\left(\frac 15 -1-5 + 25\right)\\ \\ & = \frac{96}5\end{align}$$

Answer 2

The original expression has 4 parts separated by a "$+$" or "$-$". Each of those terms can be divided by $5^x$ individually.

What is inside the parentheses in the 2nd part is what's left after dividing.

For example, $5^x/5^x=1$