The simplified value of following is
`((3)/(15) a^(5) b^(6) c^(3) xx (5)/( 9) ab^(5) c^(4) ) / (10)/(27) a^(2) bc^(3)`

Options

A. `( 9 a^(2) bc^(4))/( 10)`
B. `( 3ab ^(4) c^(3))/( 10)`
C. ` ( 3 a ^(4) b^(10) c^(4))/( 10)`
D. ` (1 a ^(4) b^(4) c^(10))/( 10)`

Answer

To simplify the expression \(\frac{\left(\frac{3}{15} a^5 b^6 c^3 \times \frac{5}{9} a b^5 c^4\right)}{\frac{10}{27} a^2 b c^3}\), we will follow these steps:

### Step 1: Rewrite the Expression
We start by rewriting the expression clearly:
\[ \frac{\left(\frac{3}{15} a^5 b^6 c^3 \times \frac{5}{9} a b^5 c^4\right)}{\frac{10}{27} a^2 b c^3} \]

### Step 2: Simplify the Numerator
First, we simplify the numerator:
\[ \frac{3}{15} = \frac{1}{5}, \quad \text{so we have: } \frac{1}{5} a^5 b^6 c^3 \times \frac{5}{9} a b^5 c^4 \]
Now, multiply the fractions:
\[ \frac{1 \cdot 5}{5 \cdot 9} = \frac{1}{9} \]
Next, combine the powers of \(a\), \(b\), and \(c\):
\[ a^5 \cdot a = a^{5+1} = a^6, \quad b^6 \cdot b^5 = b^{6+5} = b^{11}, \quad c^3 \cdot c^4 = c^{3+4} = c^7 \]
Thus, the numerator simplifies to:
\[ \frac{1}{9} a^6 b^{11} c^7 \]

### Step 3: Rewrite the Denominator
Now, we rewrite the denominator:
\[ \frac{10}{27} a^2 b c^3 \]

### Step 4: Combine the Expression
Now we can combine the numerator and denominator:
\[ \frac{\frac{1}{9} a^6 b^{11} c^7}{\frac{10}{27} a^2 b c^3} \]

### Step 5: Simplify the Fraction
To simplify this, we multiply by the reciprocal of the denominator:
\[ = \frac{1}{9} a^6 b^{11} c^7 \times \frac{27}{10} \frac{1}{a^2 b c^3} \]
This gives us:
\[ = \frac{27}{90} a^{6-2} b^{11-1} c^{7-3} \]
Calculating the powers:
\[ = \frac{27}{90} a^4 b^{10} c^4 \]

### Step 6: Simplify the Coefficient
Now simplify \(\frac{27}{90}\):
\[ \frac{27}{90} = \frac{3}{10} \]
Thus, the final simplified expression is:
\[ \frac{3}{10} a^4 b^{10} c^4 \]

### Final Answer
The simplified value of the expression is:
\[ \frac{3}{10} a^4 b^{10} c^4 \]