Find the value of `(27)^(-2//3) + ((2^(-2//3))^(-5//3))^(-9//10)`

Options

A. `1/9`
B. `2/9`
C. `11/18`
D. 1

Answer

To solve the expression \( (27)^{-\frac{2}{3}} + \left( (2^{-\frac{2}{3}})^{-\frac{5}{3}} \right)^{-\frac{9}{10}} \), we will break it down step by step.

### Step 1: Simplify \( 27^{-\frac{2}{3}} \)

First, we recognize that \( 27 \) can be expressed as \( 3^3 \):
\[ 27 = 3^3 \]
Thus, we can rewrite the expression:
\[ (27)^{-\frac{2}{3}} = (3^3)^{-\frac{2}{3}} \]
Using the power of a power property \( (a^m)^n = a^{m \cdot n} \):
\[ (3^3)^{-\frac{2}{3}} = 3^{3 \cdot -\frac{2}{3}} = 3^{-2} \]

### Step 2: Calculate \( 3^{-2} \)

Now we compute \( 3^{-2} \):
\[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \]

### Step 3: Simplify \( \left( (2^{-\frac{2}{3}})^{-\frac{5}{3}} \right)^{-\frac{9}{10}} \)

Next, we simplify the second part of the expression:
\[ (2^{-\frac{2}{3}})^{-\frac{5}{3}} = 2^{-\frac{2}{3} \cdot -\frac{5}{3}} = 2^{\frac{10}{9}} \]
Now we take this result and raise it to the power of \( -\frac{9}{10} \):
\[ (2^{\frac{10}{9}})^{-\frac{9}{10}} = 2^{\frac{10}{9} \cdot -\frac{9}{10}} = 2^{-1} \]

### Step 4: Calculate \( 2^{-1} \)

Now we compute \( 2^{-1} \):
\[ 2^{-1} = \frac{1}{2} \]

### Step 5: Combine the results

Now we combine the results from Step 2 and Step 4:
\[ \frac{1}{9} + \frac{1}{2} \]

### Step 6: Find a common denominator

To add these fractions, we need a common denominator. The least common multiple of 9 and 2 is 18:
\[ \frac{1}{9} = \frac{2}{18}, \quad \frac{1}{2} = \frac{9}{18} \]
Now we can add:
\[ \frac{2}{18} + \frac{9}{18} = \frac{11}{18} \]

### Final Answer

Thus, the value of the expression is:
\[ \frac{11}{18} \]

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