Simplifying the Expression: (a/2)^4 (8a^5)^2/a^1a^10
This article will guide you through simplifying the expression (a/2)^4 (8a^5)^2/a^1a^10.
Understanding the Rules of Exponents
To simplify this expression, we'll need to apply several exponent rules:
 Power of a product: (ab)^n = a^n * b^n
 Power of a quotient: (a/b)^n = a^n/b^n
 Product of powers: a^m * a^n = a^(m+n)
 Quotient of powers: a^m / a^n = a^(mn)
 Negative exponent: a^n = 1/a^n
StepbyStep Simplification
Let's break down the simplification step by step:

Apply the power of a quotient rule to (a/2)^4: (a/2)^4 = a^4 / 2^4

Apply the power of a product rule to (8a^5)^2: (8a^5)^2 = 8^2 * a^10

Combine the terms and simplify: (a^4/2^4) * (8^2 * a^10) / (a^1 * a^10)

Apply the product of powers rule for the 'a' terms in the numerator and denominator: (a^(4+10)) * (8^2) / (2^4 * a^(1+10))

Simplify further: a^14 * 64 / 16 * a^9

Apply the quotient of powers rule: a^(149) * 64/16

Simplify to get the final result: a^5 * 4
Final Result
Therefore, the simplified form of the expression (a/2)^4 (8a^5)^2/a^1a^10 is 4a^5.