2. Simplify expression: $$ \left(\frac{a-1}{a^{2}-2 a+1}+\frac{2(a-1)}{a^{2}-4}-\frac{4(a+1)}{a^{2}+a-2}+\frac{a}{a^{2}-3 a+2}\right) \div \frac{a^{3}+27}{36 a^{3}-144 a-36 a^{2}+144} $$ and calculate its value $$ a=-3 $$.

Question

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{a-1}{a^{2}-2a+1}+\frac{2\left(a-1\right)}{a^{2}-4}-\frac{4\left(a+1\right)}{a^{2}+a-2}+\frac{a}{a^{2}-3a+2}\right)\div \frac{a^{3}+27}{36a^{3}-144a-36a^{2}+144}$$
- step1: Divide the terms:
  $$\left(\frac{1}{a-1}+\frac{2\left(a-1\right)}{a^{2}-4}-\frac{4\left(a+1\right)}{a^{2}+a-2}+\frac{a}{a^{2}-3a+2}\right)\div \frac{a^{3}+27}{36a^{3}-144a-36a^{2}+144}$$
- step2: Calculate:
  $$\frac{6+2a}{\left(a-1\right)\left(a^{2}-4\right)}\div \frac{a^{3}+27}{36a^{3}-144a-36a^{2}+144}$$
- step3: Multiply by the reciprocal:
  $$\frac{6+2a}{\left(a-1\right)\left(a^{2}-4\right)}\times \frac{36a^{3}-144a-36a^{2}+144}{a^{3}+27}$$
- step4: Rewrite the expression:
  $$\frac{2\left(a+3\right)}{\left(a-1\right)\left(a^{2}-4\right)}\times \frac{36a^{3}-144a-36a^{2}+144}{\left(a+3\right)\left(a^{2}-3a+9\right)}$$
- step5: Reduce the fraction:
  $$\frac{2}{\left(a-1\right)\left(a^{2}-4\right)}\times \frac{36a^{3}-144a-36a^{2}+144}{a^{2}-3a+9}$$
- step6: Rewrite the expression:
  $$\frac{2}{\left(a-1\right)\left(a^{2}-4\right)}\times \frac{\left(a-1\right)\left(36a^{2}-144\right)}{a^{2}-3a+9}$$
- step7: Reduce the fraction:
  $$\frac{2}{a^{2}-4}\times \frac{36a^{2}-144}{a^{2}-3a+9}$$
- step8: Rewrite the expression:
  $$\frac{2}{a^{2}-4}\times \frac{36\left(a^{2}-4\right)}{a^{2}-3a+9}$$
- step9: Reduce the fraction:
  $$2\times \frac{36}{a^{2}-3a+9}$$
- step10: Multiply the terms:
  $$\frac{2\times 36}{a^{2}-3a+9}$$
- step11: Multiply the terms:
  $$\frac{72}{a^{2}-3a+9}$$
Let's simplify the expression step by step.

The expression we need to simplify is:

$$
\left(\frac{a-1}{a^{2}-2 a+1}+\frac{2(a-1)}{a^{2}-4}-\frac{4(a+1)}{a^{2}+a-2}+\frac{a}{a^{2}-3 a+2}\right) \div \frac{a^{3}+27}{36 a^{3}-144 a-36 a^{2}+144}
$$

### Step 1: Simplifying the Numerator

1. **First term**: 
   $$
   \frac{a-1}{a^{2}-2a+1} = \frac{a-1}{(a-1)^2} = \frac{1}{a-1} \quad (a \neq 1)
   $$

2. **Second term**: 
   $$
   \frac{2(a-1)}{a^{2}-4} = \frac{2(a-1)}{(a-2)(a+2)} \quad (a \neq 2, -2)
   $$

3. **Third term**: 
   $$
   \frac{-4(a+1)}{a^{2}+a-2} = \frac{-4(a+1)}{(a-1)(a+2)} \quad (a \neq 1, -2)
   $$

4. **Fourth term**: 
   $$
   \frac{a}{a^{2}-3a+2} = \frac{a}{(a-1)(a-2)} \quad (a \neq 1, 2)
   $$

### Step 2: Combine the Terms

Now we combine these fractions. The common denominator for the first three terms can be taken as $$(a-1)(a-2)(a+2)$$.

The combined expression becomes:

$$
\frac{(a-1)(a-2)(a+2) + 2(a-1)(a+2) - 4(a+1)(a-2) + a(a-1)(a+2)}{(a-1)(a-2)(a+2)}
$$

### Step 3: Simplifying the Denominator

The denominator is:

$$
\frac{a^{3}+27}{36 a^{3}-144 a-36 a^{2}+144}
$$

The numerator $$a^3 + 27$$ can be factored as $$(a + 3)(a^2 - 3a + 9)$$.

The denominator can be simplified further, but let's focus on the overall expression.

### Final Expression

After simplification, the overall expression simplifies to:

$$
\frac{72}{a^{2}-3a+9}
$$

### Step 4: Substitute $$a = -3$$

Now, substituting $$a = -3$$:

1. **Numerator**: $$72$$
2. **Denominator**: 
   $$
   (-3)^{2} - 3(-3) + 9 = 9 + 9 + 9 = 27
   $$

Thus, the value of the expression when $$a = -3$$ is:

$$
\frac{72}{27} = \frac{8}{3}
$$

### Conclusion

The simplified expression is $$\frac{72}{a^{2}-3a+9}$$ and its value at $$a = -3$$ is $$\frac{8}{3}$$.
The simplified expression is $$\frac{72}{a^{2}-3a+9}$$, and when $$a = -3$$, its value is $$\frac{8}{3}$$.

2. Simplify expression: \[ \left(\frac{a-1}{a^{2}-2 a+1}+\frac{2(a-1)}{a^{2}-4}-\frac{4(a+1)}{a^{2}+a-2}+\frac{a}{a^{2}-3 a+2}\right) \div \frac{a^{3}+27}{36 a^{3}-144 a-36 a^{2}+144} \] and calculate its value \( a=-3 \).

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