As a product of powers
(
−
8
)
6
   
(
−
8
)
2
   
(
−
8
)
3
   
(
−
8
)
4
   
(
−
8
)
1
   
(
−
2
3
)
3

To simplify the given expression as a product of powers:

\[
(-8)^6 \cdot (-8)^2 \cdot (-8)^3 \cdot (-8)^4 \cdot (-8)^1 \cdot (-2^3)^3
\]

### Step-by-step solution:

1. **Combine like terms**: We can combine the powers of \(-8\) by using the rule of exponents:

\[
a^m \cdot a^n = a^{m+n}
\]

So, for the powers of \(-8\):

\[
(-8)^6 \cdot (-8)^2 \cdot (-8)^3 \cdot (-8)^4 \cdot (-8)^1 = (-8)^{6+2+3+4+1} = (-8)^{16}
\]

2. **Simplify the second part**: Now let's handle the \((-2^3)^3\):

\[
(-2^3)^3 = (-8)^3
\]

3. **Multiply the two parts**: Now we have:

\[
(-8)^{16} \cdot (-8)^3
\]

Using the same exponent rule, we can combine these:

\[
(-8)^{16+3} = (-8)^{19}
\]

Thus, the final simplified expression is:

\[
(-8)^{19}
\]

### Do you want more details or have any questions?

Here are some related questions to explore:

1. How can we express \((-8)^{19}\) in its expanded form?
2. What is the value of \((-8)^{19}\) if you calculate it?
3. How do negative powers affect the final result when working with negative bases?
4. Can you explain how exponents distribute over multiplication and division?
5. What happens if one of the exponents was negative instead of positive?

**Tip:** When multiplying powers with the same base, simply add the exponents!

As a product of powers (-8)^6 (-8)^2 (-8)^3 (-8)^4 (-8)^1 (-2^3)^3

This detailed step-by-step solution simplifies the expression (-8)^6 * (-8)^2 * (-8)^3 * (-8)^4 * (-8)^1 * (-2^3)^3 using exponent rules. Learn how to combine like bases and apply the product of powers rule. Suitable for students in Grades 8-10.

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