Simplify: `[(2 10/27))^(-2/3) ÷ (11 1/9))^(-0.5)] +[(6.25)^(0.5) ÷ (-4)^(-1)]`

Options

A. `-8 (1/8)`
B. `8(1/8)`
C. `1(7/8)`
D. `-1(7/8)`

Answer

To simplify the expression `[(2 10/27)^(-2/3) ÷ (11 1/9)^(-0.5)] + [(6.25)^(0.5) ÷ (-4)^(-1)]`, we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert `2 10/27` and `11 1/9` into improper fractions.

- For `2 10/27`:
\[ 2 = \frac{2 \times 27 + 10}{27} = \frac{54 + 10}{27} = \frac{64}{27} \]

- For `11 1/9`:
\[ 11 = \frac{11 \times 9 + 1}{9} = \frac{99 + 1}{9} = \frac{100}{9} \]

### Step 2: Rewrite the Expression
Now rewrite the expression using these improper fractions:
\[ \left(\frac{64}{27}\right)^{-2/3} \div \left(\frac{100}{9}\right)^{-1/2} + \left(6.25\right)^{1/2} \div \left(-4\right)^{-1} \]

### Step 3: Convert Decimal to Fraction
Convert `6.25` to a fraction:
\[ 6.25 = \frac{625}{100} = \frac{25}{4} \]

### Step 4: Rewrite the Expression Again
Now the expression looks like this:
\[ \left(\frac{64}{27}\right)^{-2/3} \div \left(\frac{100}{9}\right)^{-1/2} + \left(\frac{25}{4}\right)^{1/2} \div \left(-4\right)^{-1} \]

### Step 5: Apply the Negative Exponents
Using the property \(a^{-n} = \frac{1}{a^n}\):
\[ \left(\frac{64}{27}\right)^{-2/3} = \frac{1}{\left(\frac{64}{27}\right)^{2/3}} \quad \text{and} \quad \left(\frac{100}{9}\right)^{-1/2} = \frac{1}{\left(\frac{100}{9}\right)^{1/2}} \]

### Step 6: Simplify Each Term
1. Simplify \(\left(\frac{64}{27}\right)^{2/3}\):
\[ = \frac{64^{2/3}}{27^{2/3}} = \frac{(4^3)^{2/3}}{(3^3)^{2/3}} = \frac{4^2}{3^2} = \frac{16}{9} \]

2. Simplify \(\left(\frac{100}{9}\right)^{1/2}\):
\[ = \frac{10}{3} \]

3. Now substitute back:
\[ \frac{1}{\frac{16}{9}} \div \frac{1}{\frac{10}{3}} = \frac{9}{16} \times \frac{3}{10} = \frac{27}{160} \]

### Step 7: Simplify the Second Part
1. Simplify \(\left(\frac{25}{4}\right)^{1/2}\):
\[ = \frac{5}{2} \]

2. Simplify \((-4)^{-1}\):
\[ = -\frac{1}{4} \]

3. Now substitute back:
\[ \frac{5}{2} \div -\frac{1}{4} = \frac{5}{2} \times -4 = -10 \]

### Step 8: Combine Both Parts
Now combine both parts:
\[ \frac{27}{160} - 10 \]

### Step 9: Convert -10 to a Fraction
Convert \(-10\) to a fraction with a denominator of 160:
\[ -10 = -\frac{1600}{160} \]

### Step 10: Combine the Fractions
Combine the fractions:
\[ \frac{27}{160} - \frac{1600}{160} = \frac{27 - 1600}{160} = \frac{-1573}{160} \]

### Final Answer
Thus, the simplified expression is:
\[ -\frac{1573}{160} \]