1) $$ \frac{27^{3-2 x} \cdot 9^{x-1}}{81^{2 x}} $$ 2) $$ \frac{6 \cdot 5^{x+1}-2 \cdot 5^{x-2}}{5^{x+3}} $$ 3) $$ \frac{2^{2009}-2^{2012}}{2^{2010}} $$

Question

UpStudy AI · Accepted Answer

Let's simplify each of the given expressions step by step.

---

### 1) Simplify $$ \frac{27^{3-2x} \cdot 9^{x-1}}{81^{2x}} $$

**Step 1:** Express all bases as powers of 3.
- $$ 27 = 3^3 $$
- $$ 9 = 3^2 $$
- $$ 81 = 3^4 $$

**Step 2:** Rewrite the expression using these bases.
$$
\frac{(3^3)^{3-2x} \cdot (3^2)^{x-1}}{(3^4)^{2x}} = \frac{3^{9 - 6x} \cdot 3^{2x - 2}}{3^{8x}}
$$

**Step 3:** Combine the exponents in the numerator.
$$
3^{9 - 6x + 2x - 2} = 3^{7 - 4x}
$$

**Step 4:** Subtract the exponent in the denominator.
$$
\frac{3^{7 - 4x}}{3^{8x}} = 3^{7 - 12x}
$$

**Final Answer:**
$$
3^{7 - 12x}
$$

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### 2) Simplify $$ \frac{6 \cdot 5^{x+1} - 2 \cdot 5^{x-2}}{5^{x+3}} $$

**Step 1:** Break down each term in the numerator.
$$
6 \cdot 5^{x+1} = 6 \cdot 5^x \cdot 5 = 30 \cdot 5^x
$$
$$
2 \cdot 5^{x-2} = 2 \cdot 5^x \cdot 5^{-2} = \frac{2 \cdot 5^x}{25}
$$

**Step 2:** Rewrite the entire expression.
$$
\frac{30 \cdot 5^x - \frac{2 \cdot 5^x}{25}}{5^{x+3}}
$$

**Step 3:** Factor out $$ 5^x $$ from the numerator.
$$
5^x \left(30 - \frac{2}{25}\right) = 5^x \left(\frac{750}{25} - \frac{2}{25}\right) = 5^x \cdot \frac{748}{25}
$$

**Step 4:** Simplify by dividing by $$ 5^{x+3} $$.
$$
\frac{5^x \cdot \frac{748}{25}}{5^x \cdot 5^3} = \frac{748}{25 \cdot 125} = \frac{748}{3125}
$$

**Final Answer:**
$$
\frac{748}{3125}
$$

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### 3) Simplify $$ \frac{2^{2009} - 2^{2012}}{2^{2010}} $$

**Step 1:** Factor out $$ 2^{2009} $$ from the numerator.
$$
2^{2009}(1 - 2^3) = 2^{2009}(1 - 8) = 2^{2009}(-7) = -7 \cdot 2^{2009}
$$

**Step 2:** Divide by $$ 2^{2010} $$.
$$
\frac{-7 \cdot 2^{2009}}{2^{2010}} = -7 \cdot 2^{-1} = -\frac{7}{2}
$$

**Final Answer:**
$$
-\frac{7}{2}
$$

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1. $$ 3^{7 - 12x} $$ 2. $$ \frac{748}{3125} $$ 3. $$ -\frac{7}{2} $$

Upstudy AI Solution

Answer

Solution

1) Simplify

2) Simplify

3) Simplify

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