Find the value of `((-8)/(5)a^(2)b^(2)c^(3))xx ((-3)/(4)ab^(2)c)` at `a= (1)/(5), b= -(1)/(2)` and c= 5.

Answer

To find the value of the expression \(\left(-\frac{8}{5} a^2 b^2 c^3\right) \times \left(-\frac{3}{4} a b^2 c\right)\) at \(a = \frac{1}{5}\), \(b = -\frac{1}{2}\), and \(c = 5\), we will follow these steps:

### Step 1: Substitute the values of \(a\), \(b\), and \(c\)

We start by substituting the given values into the expression.

\[ \left(-\frac{8}{5} \left(\frac{1}{5}\right)^2 \left(-\frac{1}{2}\right)^2 (5)^3\right) \times \left(-\frac{3}{4} \left(\frac{1}{5}\right) \left(-\frac{1}{2}\right)^2 (5)\right) \]

### Step 2: Calculate each part of the expression

First, calculate the first part:

1. Calculate \(a^2\):
\[ a^2 = \left(\frac{1}{5}\right)^2 = \frac{1}{25} \]

2. Calculate \(b^2\):
\[ b^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \]

3. Calculate \(c^3\):
\[ c^3 = 5^3 = 125 \]

Now substitute these values back into the first part:
\[ -\frac{8}{5} \times \frac{1}{25} \times \frac{1}{4} \times 125 \]

### Step 3: Simplify the first part

Now, simplify:
\[ -\frac{8 \times 125}{5 \times 25 \times 4} \]

Calculating the numerator:
\[ 8 \times 125 = 1000 \]

Calculating the denominator:
\[ 5 \times 25 = 125 \quad \text{and} \quad 125 \times 4 = 500 \]

So we have:
\[ -\frac{1000}{500} = -2 \]

### Step 4: Calculate the second part of the expression

Now calculate the second part:
\[ -\frac{3}{4} \times \left(\frac{1}{5}\right) \times \frac{1}{4} \times 5 \]

Simplifying this:
\[ -\frac{3 \times 5}{4 \times 5 \times 4} \]

Calculating the numerator:
\[ 3 \times 5 = 15 \]

Calculating the denominator:
\[ 4 \times 5 = 20 \quad \text{and} \quad 20 \times 4 = 80 \]

So we have:
\[ -\frac{15}{80} = -\frac{3}{16} \]

### Step 5: Combine both parts

Now we combine both parts:
\[ -2 \times -\frac{3}{16} = \frac{6}{16} = \frac{3}{8} \]

### Final Answer

The value of the expression is:
\[ \frac{3}{8} \]

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