It looks like you're working with an equation involving exponents and variables. Let's clarify the equation and solve it step by step.

### Given Equation:
\[ a \cdot a^3 = a \]

### Step 1: Simplify the left side of the equation
Using the property of exponents \((a^m \cdot a^n = a^{m+n})\), we can simplify the left side:
\[
a \cdot a^3 = a^{1+3} = a^4
\]
So the equation becomes:
\[
a^4 = a
\]

### Step 2: Solve the equation
To solve \(a^4 = a\), we can factor the equation:
\[
a^4 - a = 0
\]
Factor out \(a\):
\[
a(a^3 - 1) = 0
\]
This equation has two possible solutions:
1. \(a = 0\)
2. \(a^3 - 1 = 0\)

For the second part, \(a^3 - 1 = 0\), solve for \(a\):
\[
a^3 = 1
\]
Taking the cube root of both sides:
\[
a = 1
\]

### Final Answer:
The solutions are:
\[
a = 0 \quad \text{or} \quad a = 1
\]

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. What happens if the equation was \(a \cdot a^2 = a\)?
2. How do you solve equations with higher powers of \(a\)?
3. What are the general steps to factor polynomials?
4. Can you solve \(a^5 = a\)?
5. How would you solve \(a^3 = -1\)?

**Tip:** Always check your final answer by substituting it back into the original equation to verify it works.

Learn how to solve an equation involving exponents and variables step-by-step. Understand exponent rules, factorization techniques, and find roots of equations. Suitable for high school students studying algebra and exponentiation.

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