Calculate the following without using a calculator $$ \begin{array}{ll} ext { (a) } \frac{1^{2011}}{2011^{-1}} & ext { (b) }(0,125)^{-3} \ ext { (d) }\left(\frac{1}{4} ight)^{-3 x} imes 8^{2-2 x} & ext { (e) } \frac{8.9^{3}}{32^{2} .27} \ ext { (g) } \frac{1}{3^{-1}+2^{-2}} & \end{array} $$

Question

Calculate the following without using a calculator $$ \begin{array}{ll}	ext { (a) } \frac{1^{2011}}{2011^{-1}} & 	ext { (b) }(0,125)^{-3} \ 	ext { (d) }\left(\frac{1}{4}ight)^{-3 x} 	imes 8^{2-2 x} & 	ext { (e) } \frac{8.9^{3}}{32^{2} .27} \ 	ext { (g) } \frac{1}{3^{-1}+2^{-2}} & \end{array} $$

UpStudy AI · Accepted Answer

Let's solve each part step by step without using a calculator.

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**(a) $$ \frac{1^{2011}}{2011^{-1}} $$**

- **Calculate the numerator:** $$ 1^{2011} = 1 $$ (since any number raised to any power is itself if it's 1).
- **Calculate the denominator:** $$ 2011^{-1} = \frac{1}{2011} $$.
  
So,
$$
\frac{1^{2011}}{2011^{-1}} = \frac{1}{\frac{1}{2011}} = 2011
$$

**Answer:** $$ 2011 $$

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**(b) $$ (0.125)^{-3} $$**

- **Express 0.125 as a fraction:** $$ 0.125 = \frac{1}{8} $$.
- **Apply the negative exponent:** $$ \left(\frac{1}{8}\right)^{-3} = 8^{3} $$.
- **Calculate $$ 8^3 $$:** $$ 8 \times 8 \times 8 = 512 $$.

**Answer:** $$ 512 $$

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**(d) $$ \left(\frac{1}{4}\right)^{-3x} \times 8^{2-2x} $$**

- **Express in terms of powers of 2:**
  - $$ \frac{1}{4} = 2^{-2} $$ so $$ \left(\frac{1}{4}\right)^{-3x} = (2^{-2})^{-3x} = 2^{6x} $$.
  - $$ 8 = 2^{3} $$ so $$ 8^{2-2x} = (2^{3})^{2-2x} = 2^{6 - 6x} $$.
  
- **Multiply the exponents:** $$ 2^{6x} \times 2^{6 - 6x} = 2^{6x + 6 - 6x} = 2^{6} = 64 $$.

**Answer:** $$ 64 $$

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**(e) $$ \frac{8 \times 9^{3}}{32^{2} \times 27} $$**

- **Factorize each number:**
  - $$ 8 = 2^{3} $$
  - $$ 9 = 3^{2} $$ so $$ 9^{3} = 3^{6} $$
  - $$ 32 = 2^{5} $$ so $$ 32^{2} = 2^{10} $$
  - $$ 27 = 3^{3} $$
  
- **Substitute back into the expression:**
  $$
  \frac{2^{3} \times 3^{6}}{2^{10} \times 3^{3}} = 2^{3-10} \times 3^{6-3} = 2^{-7} \times 3^{3} = \frac{27}{128}
  $$

**Answer:** $$ \frac{27}{128} $$

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**(g) $$ \frac{1}{3^{-1} + 2^{-2}} $$**

- **Calculate each term in the denominator:**
  - $$ 3^{-1} = \frac{1}{3} $$
  - $$ 2^{-2} = \frac{1}{4} $$
  
- **Add them together:**
  $$
  \frac{1}{3} + \frac{1}{4} = \frac{4 + 3}{12} = \frac{7}{12}
  $$
  
- **Take the reciprocal:**
  $$
  \frac{1}{\frac{7}{12}} = \frac{12}{7}
  $$

**Answer:** $$ \frac{12}{7} $$

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**Summary of Answers:**
- **(a)** $$ 2011 $$
- **(b)** $$ 512 $$
- **(d)** $$ 64 $$
- **(e)** $$ \frac{27}{128} $$
- **(g)** $$ \frac{12}{7} $$
**(a) 2011** **(b) 512** **(d) 64** **(e) 27⁄128** **(g) 12⁄7**

Calculate the following without using a calculator

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