Evaluate :
`5^(-4)xx(125)^((5)/(3))+(25)^(-(1)/(2))`

Answer

To evaluate the expression \( 5^{-4} \times (125)^{\frac{5}{3}} + (25)^{-\frac{1}{2}} \), we will follow these steps:

### Step 1: Rewrite the bases in terms of 5
We know that:
- \( 125 = 5^3 \)
- \( 25 = 5^2 \)

Thus, we can rewrite the expression as:
\[ 5^{-4} \times (5^3)^{\frac{5}{3}} + (5^2)^{-\frac{1}{2}} \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we can simplify:
\[ (5^3)^{\frac{5}{3}} = 5^{3 \cdot \frac{5}{3}} = 5^5 \]
And,
\[ (5^2)^{-\frac{1}{2}} = 5^{2 \cdot -\frac{1}{2}} = 5^{-1} \]

So now our expression becomes:
\[ 5^{-4} \times 5^5 + 5^{-1} \]

### Step 3: Combine the terms
Using the property \( a^m \times a^n = a^{m+n} \):
\[ 5^{-4} \times 5^5 = 5^{-4 + 5} = 5^1 = 5 \]

Now, we can rewrite the expression:
\[ 5 + 5^{-1} \]

### Step 4: Simplify \( 5^{-1} \)
We know that:
\[ 5^{-1} = \frac{1}{5} \]

Thus, the expression now is:
\[ 5 + \frac{1}{5} \]

### Step 5: Find a common denominator
To add these two terms, we convert \( 5 \) into a fraction:
\[ 5 = \frac{25}{5} \]

Now we can add:
\[ \frac{25}{5} + \frac{1}{5} = \frac{25 + 1}{5} = \frac{26}{5} \]

### Final Answer
Thus, the evaluated expression is:
\[ \frac{26}{5} \]

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