To simplify the expression [tex]\(\left(2 a^6 b^9 c^9\right)^5\)[/tex], we'll apply the laws of exponents systematically.
### Step-by-Step Solution:
1. Apply the Power of a Product Rule:
The rule states that [tex]\((xy)^n = x^n \cdot y^n\)[/tex]. Thus, we distribute the power of 5 to each term inside the parentheses:
[tex]\[
\left(2 a^6 b^9 c^9\right)^5 = 2^5 \cdot (a^6)^5 \cdot (b^9)^5 \cdot (c^9)^5
\][/tex]
2. Simplify Each Term:
- First, simplify the constant term: [tex]\(2^5\)[/tex].
[tex]\[
2^5 = 32
\][/tex]
- Next, use the power of a power rule for the variable terms: [tex]\((x^m)^n = x^{m \cdot n}\)[/tex].
- For [tex]\(a^6\)[/tex]:
[tex]\[
(a^6)^5 = a^{6 \cdot 5} = a^{30}
\][/tex]
- For [tex]\(b^9\)[/tex]:
[tex]\[
(b^9)^5 = b^{9 \cdot 5} = b^{45}
\][/tex]
- For [tex]\(c^9\)[/tex]:
[tex]\[
(c^9)^5 = c^{9 \cdot 5} = c^{45}
\][/tex]
3. Combine All the Simplified Terms:
Putting all the simplified terms together, we get:
[tex]\[
32 \cdot a^{30} \cdot b^{45} \cdot c^{45}
\][/tex]
So, the simplified expression is:
[tex]\[
32 \cdot a^{30} \cdot b^{45} \cdot c^{45}
\][/tex]
