Sure, let's simplify the expression step by step:
Given expression:
[tex]\[ \frac{(x^a)^8 \cdot (x^2)^5 \cdot (x^2)^{-4}}{(x^2)^5} \][/tex]
Step 1: Use the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex] to simplify.
[tex]\[ (x^a)^8 = x^{8a} \][/tex]
[tex]\[ (x^2)^5 = x^{10} \][/tex]
[tex]\[ (x^2)^{-4} = x^{-8} \][/tex]
The expression now looks like:
[tex]\[ \frac{x^{8a} \cdot x^{10} \cdot x^{-8}}{x^{10}} \][/tex]
Step 2: Apply the product of powers rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
[tex]\[ x^{8a} \cdot x^{10} \cdot x^{-8} = x^{8a + 10 - 8} \][/tex]
[tex]\[ x^{8a + 10 - 8} = x^{8a + 2} \][/tex]
The expression now looks like:
[tex]\[ \frac{x^{8a + 2}}{x^{10}} \][/tex]
Step 3: Apply the quotient of powers rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
[tex]\[ \frac{x^{8a + 2}}{x^{10}} = x^{(8a + 2) - 10} \][/tex]
[tex]\[ x^{8a + 2 - 10} = x^{8a - 8} \][/tex]
So, the simplified form of the given expression is:
[tex]\[ x^{8a - 8} \][/tex]
