Simplify the following expressions and evaluate for the given values of 'a': 1. (a^7)^3 / a^18, for a = 2 2. a^5 / a^16, for a = 2 3. a^4 / a^5, for a = 3

Let's break down each expression and solve it step-by-step!

1. Expression 1: \( \frac{(a^7)^3}{a^{18}} \) for \( a = 2 \)
   1. First, simplify the numerator using the power of a power rule: \( (a^7)^3 = a^{7 \times 3} = a^{21} \)
   2. Now the expression is: \( \frac{a^{21}}{a^{18}} \)
   3. Using the rule for dividing powers with the same base, we subtract the exponents: \( a^{21 - 18} = a^3 \)
   4. Finally, substitute \( a = 2 \): \( 2^3 = 2 \times 2 \times 2 = 8 \)
2. Expression 2: \( \frac{a^5}{a^{16}} \) for \( a = 2 \)
   1. Using the rule for dividing powers with the same base, subtract the exponents: \( a^{5 - 16} = a^{-11} \)
   2. A negative exponent means we take the reciprocal: \( \frac{1}{a^{11}} \)
   3. Substitute \( a = 2 \): \( \frac{1}{2^{11}} = \frac{1}{2048} \)
3. Expression 3: \( \frac{a^4}{a^5} \) for \( a = 3 \)
   1. Using the rule for dividing powers with the same base, subtract the exponents: \( a^{4 - 5} = a^{-1} \)
   2. A negative exponent means we take the reciprocal: \( \frac{1}{a} \)
   3. Substitute \( a = 3 \): \( \frac{1}{3} \)

Summary of answers:

• For \( \frac{(a^7)^3}{a^{18}} \) when \( a = 2 \), the answer is 8.
• For \( \frac{a^5}{a^{16}} \) when \( a = 2 \), the answer is \( \frac{1}{2048} \).
• For \( \frac{a^4}{a^5} \) when \( a = 3 \), the answer is \( \frac{1}{3} \).