Simplify:
`(625)/(15625) xx 5^2 xx sqrt(5^(-10)) div root(3)(5/(5^(16)))`

Options

A. a) 1
B. b) 0
C. c) 0.5
D. d) 0.25

Answer

To simplify the expression `(625)/(15625) * 5^2 * sqrt(5^(-10)) / root(3)(5/(5^(16)))`, we will follow these steps:

### Step 1: Simplify the fraction
We start with the fraction \( \frac{625}{15625} \).

- \( 625 = 5^4 \) (since \( 5^4 = 625 \))
- \( 15625 = 5^6 \) (since \( 5^6 = 15625 \))

Thus, we can rewrite the fraction as:
\[ \frac{625}{15625} = \frac{5^4}{5^6} = 5^{4-6} = 5^{-2} \]

**Hint for Step 1:** Remember that \( \frac{a^m}{a^n} = a^{m-n} \).

### Step 2: Simplify \( 5^2 \)
Next, we have \( 5^2 \) which remains as it is.

**Hint for Step 2:** Just keep the powers of 5 as they are unless you need to combine them.

### Step 3: Simplify \( \sqrt{5^{-10}} \)
Now we simplify \( \sqrt{5^{-10}} \):
\[ \sqrt{5^{-10}} = 5^{-10/2} = 5^{-5} \]

**Hint for Step 3:** Remember that \( \sqrt{a^m} = a^{m/2} \).

### Step 4: Simplify the cubic root
Next, we simplify \( \text{root}(3)\left(\frac{5}{5^{16}}\right) \):
\[ \frac{5}{5^{16}} = 5^{1-16} = 5^{-15} \]
Now applying the cubic root:
\[ \text{root}(3)(5^{-15}) = 5^{-15/3} = 5^{-5} \]

**Hint for Step 4:** For cubic roots, use \( \text{root}(3)(a^m) = a^{m/3} \).

### Step 5: Combine all parts
Now we combine all the parts we have simplified:
\[ 5^{-2} \cdot 5^2 \cdot 5^{-5} \div 5^{-5} \]

### Step 6: Handle the division
Using the property of exponents:
\[ 5^{-2} \cdot 5^2 \cdot 5^{-5} \cdot 5^{5} = 5^{-2 + 2 - 5 + 5} = 5^{0} = 1 \]

**Hint for Step 6:** When dividing or multiplying powers of the same base, add or subtract their exponents accordingly.

### Final Answer
The simplified expression is:
\[ \boxed{1} \]