`|{:(5sqrt(log_(5)3),,5sqrt(log_(5)3),,5sqrt(log_(5)3)),(3^(-log_(1//3)4),,(0.1)^(log_(0.01)4),,7^(log_(7)3)),(7,,3,,5):}|" is equal to ""____"`

Explanation

To solve the determinant

\[ D = \begin{vmatrix} 5\sqrt{\log_{5}3} & 5\sqrt{\log_{5}3} & 5\sqrt{\log_{5}3} \\ 3^{-\log_{1/3}4} & (0.1)^{\log_{0.01}4} & 7^{\log_{7}3} \\ 7 & 3 & 5 \end{vmatrix} \]

we will simplify the expressions in the determinant and then calculate its value.

### Step 1: Simplify the elements of the determinant

1. **First Row Elements**:
- The first row consists of the same element \(5\sqrt{\log_{5}3}\).

2. **Second Row Elements**:
- For \(3^{-\log_{1/3}4}\):
\[ 3^{-\log_{1/3}4} = 3^{\log_{3}4} = 4 \]
- For \((0.1)^{\log_{0.01}4}\):
\[ 0.1 = 10^{-1} \quad \text{and} \quad 0.01 = 10^{-2} \implies \log_{0.01}4 = \frac{\log_{10}4}{\log_{10}0.01} = \frac{\log_{10}4}{-2} = -\frac{1}{2}\log_{10}4 \]
\[ (0.1)^{\log_{0.01}4} = (10^{-1})^{-\frac{1}{2}\log_{10}4} = 10^{\frac{1}{2}\log_{10}4} = 4^{1/2} = 2 \]
- For \(7^{\log_{7}3}\):
\[ 7^{\log_{7}3} = 3 \]

3. **Second Row After Simplification**:
- The second row becomes \([4, 2, 3]\).

### Step 2: Rewrite the determinant

Now, the determinant can be rewritten as:

\[ D = \begin{vmatrix} 5\sqrt{\log_{5}3} & 5\sqrt{\log_{5}3} & 5\sqrt{\log_{5}3} \\ 4 & 2 & 3 \\ 7 & 3 & 5 \end{vmatrix} \]

### Step 3: Apply column operations

We can simplify the determinant by performing column operations:

- Let \(C_2 \to C_2 - C_1\)
- Let \(C_3 \to C_3 - C_1\)

This gives us:

\[ D = \begin{vmatrix} 5\sqrt{\log_{5}3} & 0 & 0 \\ 4 & -2 & -1 \\ 7 & -4 & -2 \end{vmatrix} \]

### Step 4: Calculate the determinant

Now we can calculate the determinant using the first column:

\[ D = 5\sqrt{\log_{5}3} \cdot \begin{vmatrix} -2 & -1 \\ -4 & -2 \end{vmatrix} \]

Calculating the \(2 \times 2\) determinant:

\[ \begin{vmatrix} -2 & -1 \\ -4 & -2 \end{vmatrix} = (-2)(-2) - (-1)(-4) = 4 - 4 = 0 \]

### Step 5: Conclusion

Thus, the value of the determinant \(D\) is:

\[ D = 5\sqrt{\log_{5}3} \cdot 0 = 0 \]

### Final Answer

The value of the determinant is \(0\).

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