5 Write the following in index form \( \begin{array}{ll}\text { a) } 2 \times a \times b \times a \times 2 \times a \times 2 \times b & \text { b) } 3 a \times 2 b \times 3 a \times 2 b \\ \text { c) } 4 \mathrm{~d} \times 2 \mathrm{f} \times 4 \mathrm{~d} \times 2 \mathrm{f} & \text { d) } 5 \mathrm{~b} \text { (5ab) } 5 \mathrm{~b} \\ \text { e) } 2 \mathrm{~cd}(2 \mathrm{~cd})(2 \mathrm{~cd})(2 \mathrm{~cd}) & \text { f) } 7 \mathrm{ab} \times 7 \mathrm{ab} \times \mathrm{ab} \times 7\end{array} \)

Certainly! Let's rewrite each expression in index form by combining like terms and using exponents. ### a) \( 2 \times a \times b \times a \times 2 \times a \times 2 \times b \) 1. **Combine the constants:** \[ 2 \times 2 \times 2 = 8 \] 2. **Combine the variables:** \[ a \times a \times a = a^3 \\ b \times b = b^2 \] 3. **Combine everything:** \[ 8a^3b^2 \] ### b) \( 3a \times 2b \times 3a \times 2b \) 1. **Combine the constants:** \[ 3 \times 2 \times 3 \times 2 = 36 \] 2. **Combine the variables:** \[ a \times a = a^2 \\ b \times b = b^2 \] 3. **Combine everything:** \[ 36a^2b^2 \] ### c) \( 4d \times 2f \times 4d \times 2f \) 1. **Combine the constants:** \[ 4 \times 2 \times 4 \times 2 = 64 \] 2. **Combine the variables:** \[ d \times d = d^2 \\ f \times f = f^2 \] 3. **Combine everything:** \[ 64d^2f^2 \] ### d) \( 5b \times (5ab) \times 5b \) 1. **Expand the expression:** \[ 5b \times 5ab \times 5b \] 2. **Combine the constants:** \[ 5 \times 5 \times 5 = 125 \] 3. **Combine the variables:** \[ b \times ab \times b = a \times b^3 \] 4. **Combine everything:** \[ 125ab^3 \] ### e) \( 2cd \times (2cd) \times (2cd) \times (2cd) \) 1. **Expand the expression:** \[ 2cd \times 2cd \times 2cd \times 2cd \] 2. **Combine the constants:** \[ 2 \times 2 \times 2 \times 2 = 16 \] 3. **Combine the variables:** \[ c \times c \times c \times c = c^4 \\ d \times d \times d \times d = d^4 \] 4. **Combine everything:** \[ 16c^4d^4 \] ### f) \( 7ab \times 7ab \times ab \times 7 \) 1. **Combine the constants:** \[ 7 \times 7 \times 7 = 343 \] 2. **Combine the variables:** \[ ab \times ab \times ab = a^3b^3 \] 3. **Combine everything:** \[ 343a^3b^3 \] --- **Summary of Index Forms:** a) \( 8a^3b^2 \) b) \( 36a^2b^2 \) c) \( 64d^2f^2 \) d) \( 125ab^3 \) e) \( 16c^4d^4 \) f) \( 343a^3b^3 \)