Sure, let's simplify the expression [tex]\(\left(x^2\right)^5\left(x^5\right)^2\)[/tex] using the rules of exponents step-by-step.
### Step 1: Apply the Power of a Power Rule
First, we need to simplify each part of the expression [tex]\(\left(x^2\right)^5\)[/tex] and [tex]\(\left(x^5\right)^2\)[/tex] using the power of a power rule which states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
1. Simplify [tex]\(\left(x^2\right)^5\)[/tex]:
[tex]\[
(x^2)^5 = x^{2 \cdot 5} = x^{10}
\][/tex]
2. Simplify [tex]\(\left(x^5\right)^2\)[/tex]:
[tex]\[
(x^5)^2 = x^{5 \cdot 2} = x^{10}
\][/tex]
### Step 2: Apply the Product of Powers Rule
Now that we have simplified the expression to [tex]\(x^{10} \cdot x^{10}\)[/tex], we use the product of powers rule which states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
Combine the exponents:
[tex]\[
x^{10} \cdot x^{10} = x^{10+10} = x^{20}
\][/tex]
### Conclusion
Thus, the simplified expression is:
[tex]\[
\left(x^2\right)^5\left(x^5\right)^2 = x^{20}
\][/tex]
So, the simplified expression is [tex]\(x^{20}\)[/tex].
