\( \frac { 18 ^ { d } \cdot 15 ^ { a + 1 } } { 30 ^ { d - 1 } } \times \frac { ( 3 ^ { a } , 6 ) ^ { 2 } } { 9 ^ { a } - 1 } \div \frac { 36 ^ { a + 1 } } { 18 ^ { d - 1 } } \)

Let's simplify the expression step by step: 1. **First Expression**: \( \frac { 18 ^ { d } \cdot 15 ^ { a + 1 } } { 30 ^ { d - 1 } } \) 2. **Second Expression**: \( \frac { ( 3 ^ { a } , 6 ) ^ { 2 } } { 9 ^ { a } - 1 } \) 3. **Third Expression**: \( \frac { 36 ^ { a + 1 } } { 18 ^ { d - 1 } } \) We will first simplify each part and then combine them according to the operations given (multiplication and division). ### Step 1: Simplifying Each Expression **First Expression**: \[ \frac { 18 ^ { d } \cdot 15 ^ { a + 1 } } { 30 ^ { d - 1 } } \] **Second Expression**: \[ \frac { ( 3 ^ { a } , 6 ) ^ { 2 } } { 9 ^ { a } - 1 } \] **Third Expression**: \[ \frac { 36 ^ { a + 1 } } { 18 ^ { d - 1 } } \] ### Step 2: Combine the Expressions The overall expression can be rewritten as: \[ \left( \frac { 18 ^ { d } \cdot 15 ^ { a + 1 } } { 30 ^ { d - 1 } } \times \frac { ( 3 ^ { a } , 6 ) ^ { 2 } } { 9 ^ { a } - 1 } \right) \div \frac { 36 ^ { a + 1 } } { 18 ^ { d - 1 } } \] This can be expressed as: \[ \left( \frac { 18 ^ { d } \cdot 15 ^ { a + 1 } \cdot ( 3 ^ { a } , 6 ) ^ { 2 } } { 30 ^ { d - 1 } \cdot ( 9 ^ { a } - 1 ) } \right) \cdot \frac { 18 ^ { d - 1 } } { 36 ^ { a + 1 } } \] Now, let's simplify this entire expression. ### Step 3: Perform the Simplification I will now perform the simplification of the entire expression. It seems that the expression is too complex for direct simplification in one step. Let's break it down further into manageable parts and simplify each part step by step. ### Step 1: Simplify Each Component 1. **First Expression**: \[ \frac { 18 ^ { d } \cdot 15 ^ { a + 1 } } { 30 ^ { d - 1 } } \] 2. **Second Expression**: \[ \frac { ( 3 ^ { a } , 6 ) ^ { 2 } } { 9 ^ { a } - 1 } \] 3. **Third Expression**: \[ \frac { 36 ^ { a + 1 } } { 18 ^ { d - 1 } } \] ### Step 2: Simplify Each Expression Individually #### First Expression Let's simplify \( \frac { 18 ^ { d } \cdot 15 ^ { a + 1 } } { 30 ^ { d - 1 } } \). #### Second Expression Next, we will simplify \( \frac { ( 3 ^ { a } , 6 ) ^ { 2 } } { 9 ^ { a } - 1 } \). #### Third Expression Finally, we will simplify \( \frac { 36 ^ { a + 1 } } { 18 ^ { d - 1 } } \). I will perform these simplifications one by one. Let's start with the first expression. Simplify the expression by following steps: - step0: Solution: \(\frac{18^{d}\times 15^{a+1}}{30^{d-1}}\) - step1: Factor the expression: \(\frac{6^{d}\times 3^{d}\times 15^{a+1}}{6^{d-1}\times 5^{d-1}}\) - step2: Reduce the fraction: \(\frac{6\times 3^{d}\times 15^{a+1}}{5^{d-1}}\) - step3: Simplify: \(\frac{6\times 3^{d+a+1}\times 5^{a+1}}{5^{d-1}}\) - step4: Reduce the fraction: \(6\times 3^{d+a+1}\times 5^{a+2-d}\) Calculate or simplify the expression \( (3^a * 6)^2 \). Simplify the expression by following steps: - step0: Solution: \(\left(3^{a}\times 6\right)^{2}\) - step1: Multiply the terms: \(\left(6\times 3^{a}\right)^{2}\) - step2: Evaluate the power: \(6^{2}\left(3^{a}\right)^{2}\) - step3: Evaluate the power: \(36\left(3^{a}\right)^{2}\) - step4: Evaluate the power: \(36\times 3^{2a}\) Calculate or simplify the expression \( 9^a - 1 \). Simplify the expression by following steps: - step0: Solution: \(9^{a}-1\) Calculate or simplify the expression \( \frac { 36^{a + 1} } { 18^{d - 1} } \). Simplify the expression by following steps: - step0: Solution: \(\frac{36^{a+1}}{18^{d-1}}\) - step1: Calculate: \(\frac{2^{3+2a}\times 9^{a+2}\times \frac{1}{18}}{18^{d}\times \frac{1}{18}}\) - step2: Reduce the fraction: \(\frac{2^{3+2a}\times 9^{a+2}}{18^{d}}\) Now that we have simplified each component, let's summarize the results: 1. **First Expression**: \[ \frac { 18 ^ { d } \cdot 15 ^ { a + 1 } } { 30 ^ { d - 1 } } = 6 \times 3^{d+a+1} \times 5^{a+2-d} \] 2. **Second Expression**: \[ \frac { ( 3 ^ { a } , 6 ) ^ { 2 } } { 9 ^ { a } - 1 } = \frac{36 \times 3^{2a}}{9^{a}-1} \] 3. **Third Expression**: \[ \frac { 36^{a + 1} } { 18^{d - 1} } = \frac{2^{3+2a} \times 9^{a+2}}{18^{d}} \] ### Step 3: Combine the Expressions Now we can combine these results into the overall expression: \[ \left( 6 \times 3^{d+a+1} \times 5^{a+2-d} \times \frac{36 \times 3^{2a}}{9^{a}-1} \right) \cdot \frac{2^{3+2a} \times 9^{a+2}}{18^{d}} \] ### Step 4: Simplify the Combined Expression Let's simplify this combined expression. I will perform the simplification now. Simplify the expression by following steps: - step0: Solution: \(\left(6\times 3^{d+a+1}\times 5^{a+2-d}\times \frac{36\times 3^{2a}}{9^{a}-1}\right)\times \frac{2^{3+2a}\times 9^{a+2}}{18^{d}}\) - step1: Remove the parentheses: \(6\times 3^{d+a+1}\times 5^{a+2-d}\times \frac{36\times 3^{2a}}{9^{a}-1}\times \frac{2^{3+2a}\times 9^{a+2}}{18^{d}}\) - step2: Multiply the terms: \(\frac{216\times 3^{d+3a+1}\times 5^{a+2-d}}{9^{a}-1}\times \frac{2^{3+2a}\times 9^{a+2}}{18^{d}}\) - step3: Rewrite the expression: \(\frac{216\times 3^{d+3a+1}\times 5^{a+2-d}}{9^{a}-1}\times \frac{2^{3+2a}\times 9^{a+2}}{3^{d}\times 6^{d}}\) - step4: Reduce the fraction: \(\frac{216\times 3^{3a+1}\times 5^{a+2-d}}{9^{a}-1}\times \frac{2^{3+2a}\times 9^{a+2}}{6^{d}}\) - step5: Multiply the terms: \(\frac{216\times 3^{3a+1}\times 5^{a+2-d}\times 2^{3+2a}\times 9^{a+2}}{\left(9^{a}-1\right)\times 6^{d}}\) - step6: Factor the expression: \(\frac{6^{3}\times 3^{3a+1}\times 5^{a+2-d}\times 2^{3+2a}\times 9^{a+2}}{\left(9^{a}-1\right)\times 6^{d}}\) - step7: Divide the numbers: \(\frac{6^{3-d}\times 3^{3a+1}\times 5^{a+2-d}\times 2^{3+2a}\times 9^{a+2}}{9^{a}-1}\) - step8: Simplify: \(\frac{2^{6-d+2a}\times 3^{8-d+5a}\times 5^{a+2-d}}{9^{a}-1}\) The final simplified expression is: \[ \frac{2^{6-d+2a} \times 3^{8-d+5a} \times 5^{a+2-d}}{9^{a}-1} \] This represents the result of the original expression you provided.