Simplify the following without the use of a calculator: $$ \begin{array}{lll}7.27^{-\frac{1}{3}} \cdot \underbrace{81^{-\frac{1}{4}}} \cdot 9^{2} & ext { 8. }\left(\frac{36}{64} ight)^{-\frac{3}{2}} & ext { 9. }(2,25)^{-\frac{1}{2}}\end{array} $$

Question

Simplify the following without the use of a calculator: $$ \begin{array}{lll}7.27^{-\frac{1}{3}} \cdot \underbrace{81^{-\frac{1}{4}}} \cdot 9^{2} & 	ext { 8. }\left(\frac{36}{64}ight)^{-\frac{3}{2}} & 	ext { 9. }(2,25)^{-\frac{1}{2}}\end{array} $$

UpStudy AI · Accepted Answer

Sure, let's simplify each of the expressions step by step without using a calculator.

### 7. $$27^{-\frac{1}{3}} \cdot 81^{-\frac{1}{4}} \cdot 9^{2}$$

1. **Simplify each term individually:**
   - $$27^{-\frac{1}{3}}$$:
     - $$27 = 3^3$$
     - $$27^{-\frac{1}{3}} = (3^3)^{-\frac{1}{3}} = 3^{-1} = \frac{1}{3}$$

- $$81^{-\frac{1}{4}}$$:
     - $$81 = 3^4$$
     - $$81^{-\frac{1}{4}} = (3^4)^{-\frac{1}{4}} = 3^{-1} = \frac{1}{3}$$

- $$9^{2}$$:
     - $$9 = 3^2$$
     - $$9^{2} = (3^2)^2 = 3^4 = 81$$

2. **Multiply the simplified terms:**
   $$
   \frac{1}{3} \cdot \frac{1}{3} \cdot 81 = \frac{1}{9} \cdot 81 = 9
   $$

**Simplified Result:** $$9$$

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### 8. $$\left(\frac{36}{64}\right)^{-\frac{3}{2}}$$

1. **Simplify the fraction inside the exponent:**
   $$
   \frac{36}{64} = \frac{9}{16}
   $$

2. **Apply the negative exponent:**
   $$
   \left(\frac{9}{16}\right)^{-\frac{3}{2}} = \left(\frac{16}{9}\right)^{\frac{3}{2}}
   $$

3. **Simplify the exponent:**
   $$
   \left(\frac{16}{9}\right)^{\frac{3}{2}} = \left(\sqrt{\frac{16}{9}}\right)^3 = \left(\frac{4}{3}\right)^3 = \frac{64}{27}
   $$

**Simplified Result:** $$\frac{64}{27}$$

---

### 9. $$(2.25)^{-\frac{1}{2}}$$

1. **Convert the decimal to a fraction:**
   $$
   2.25 = \frac{9}{4}
   $$

2. **Apply the negative exponent:**
   $$
   \left(\frac{9}{4}\right)^{-\frac{1}{2}} = \left(\frac{4}{9}\right)^{\frac{1}{2}} = \sqrt{\frac{4}{9}} = \frac{2}{3}
   $$

**Simplified Result:** $$\frac{2}{3}$$

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**Summary of Simplified Expressions:**

1. $$27^{-\frac{1}{3}} \cdot 81^{-\frac{1}{4}} \cdot 9^{2} = 9$$
2. $$\left(\frac{36}{64}\right)^{-\frac{3}{2}} = \frac{64}{27}$$
3. $$(2.25)^{-\frac{1}{2}} = \frac{2}{3}$$
The simplified results are: 1. $$9$$ 2. $$\frac{64}{27}$$ 3. $$\frac{2}{3}$$

Simplify the following without the use of a calculator:

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