Simplify each of the folloiwing and express with positive index :
`(32)^(-(2)/(5))+(125)^(-(2)/(3))`

Answer

To simplify the expression \( (32)^{-\frac{2}{5}} + (125)^{-\frac{2}{3}} \) and express it with positive indices, follow these steps:

### Step 1: Rewrite the bases in terms of prime factors
We can express 32 and 125 in terms of their prime factors:
- \( 32 = 2^5 \)
- \( 125 = 5^3 \)

So, we can rewrite the expression as:
\[ (2^5)^{-\frac{2}{5}} + (5^3)^{-\frac{2}{3}} \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we simplify each term:
\[ (2^5)^{-\frac{2}{5}} = 2^{5 \cdot -\frac{2}{5}} = 2^{-2} \]
\[ (5^3)^{-\frac{2}{3}} = 5^{3 \cdot -\frac{2}{3}} = 5^{-2} \]

Now, our expression becomes:
\[ 2^{-2} + 5^{-2} \]

### Step 3: Rewrite with positive indices
To express the terms with positive indices, we use the property \( a^{-n} = \frac{1}{a^n} \):
\[ 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \]
\[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

Thus, the expression now is:
\[ \frac{1}{4} + \frac{1}{25} \]

### Step 4: Find a common denominator
The least common multiple (LCM) of 4 and 25 is 100. We convert each fraction:
\[ \frac{1}{4} = \frac{25}{100} \]
\[ \frac{1}{25} = \frac{4}{100} \]

### Step 5: Add the fractions
Now we can add the fractions:
\[ \frac{25}{100} + \frac{4}{100} = \frac{25 + 4}{100} = \frac{29}{100} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{29}{100} \]

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