**Simplifying Algebraic Expressions: A Step-by-Step Guide**

In this article, we will explore the simplification of the algebraic expression `(-2a)^3(2a)^-2/3`

under the condition `a ≠ 0`

. We will follow a step-by-step approach to break down the expression and simplify it to its most basic form.

**Step 1: Understand the Expression**

The given expression is `(-2a)^3(2a)^-2/3`

. This expression involves two terms: `(-2a)^3`

and `(2a)^-2/3`

. Our goal is to simplify this expression by combining these two terms.

**Step 2: Simplify the First Term**

Let's start by simplifying the first term `(-2a)^3`

. Using the power rule of exponents, which states that `(ab)^n = a^n b^n`

, we can rewrite the term as:

`(-2a)^3 = (-2)^3 a^3`

Now, we can simplify the term by evaluating the cube of `-2`

:

`(-2)^3 = -8`

So, the simplified first term is `(-2a)^3 = -8a^3`

.

**Step 3: Simplify the Second Term**

Next, let's simplify the second term `(2a)^-2/3`

. To simplify this term, we need to recall the rules of exponentiation. Specifically, we need to use the rule that states `a^(-n) = 1/a^n`

.

Using this rule, we can rewrite the term as:

`(2a)^-2/3 = 1/(2a)^(2/3)`

Now, we can simplify the term further by evaluating the expression inside the parentheses:

`1/(2a)^(2/3) = 1/(2^(2/3) a^(2/3))`

**Step 4: Combine the Terms**

Now that we have simplified both terms, we can combine them to get the final expression. Recall that the original expression is a product of the two terms:

`(-2a)^3(2a)^-2/3`

Substituting the simplified terms, we get:

`-8a^3(1/(2^(2/3) a^(2/3)))`

**Step 5: Simplify the Final Expression**

To simplify the final expression, we can combine the terms by canceling out any common factors. Notice that `a^3`

and `a^(2/3)`

have a common factor of `a^(2/3)`

. We can cancel out this term to get:

`-8a^(3 - 2/3)/(2^(2/3))`

Now, we can simplify the expression further by evaluating the exponents:

`-8a^(7/3)/(2^(2/3))`

Finally, we can simplify the expression to its most basic form:

`-64a^(7/3)/4`

Simplifying further, we get:

`-16a^(7/3)`

And that's the final answer!

**Conclusion**

In this article, we simplified the algebraic expression `(-2a)^3(2a)^-2/3`

under the condition `a ≠ 0`

. By following a step-by-step approach, we broke down the expression into its component parts, simplified each term, and combined them to get the final answer. The simplified expression is `-16a^(7/3)`

.